## Main

The perturbative effect of a structured vacuum is the renowned Purcell effect, which states that the lifetime of an atom in such a space will be proportional to the local photonic density of states (DOS) near the atomic transition frequency. In practice, the birth of the photonic crystal, which greatly modifies the vacuum fluctuations, has enabled the control of spontaneous emission of various emitters, such as quantum dots5,6, magnons7 and superconducting qubits8. However, when an atom is strongly coupled to a photonic crystal, non-perturbative effects become important, and significantly enrich the physics. For instance, a single-photon bound state has been predicted to emerge within the gap9, and spontaneous emission of the atom will thus exhibit Rabi oscillation and light-trapping behaviour. In contrast to electronic bandgap systems, even multiple photons can be simultaneously localized by a single atom, and the coherent photonic transport within the otherwise forbidden bandgap can have a strongly correlated nature10,11,12. In contrast to a system with discrete cavity modes, which is well described by the single-mode or multimode Jaynes–Cummings Hamiltonian13,14, a continuous density of states enables the formation of a localized state in the bandgap. While other spin-boson problems with continuous DOS have also been studied experimentally15,16 or theoretically17,18 with superconducting circuits, this work explores physics near the band edge, where localized states emerge and reservoir engineering becomes possible.

Light–matter interactions are being actively pursued using cold atoms coupled to optical photonic crystals19,20, where the study of photonic band edge effects requires a combination of challenging nanostructure fabrication and optical laser trapping. Although impressive progress has been made, atoms are only weakly coupled to photonic crystal waveguides20, potentially limiting the physics to the perturbative regime. In this letter, using a microwave photonic crystal and a superconducting transmon qubit, we are able to reach the strong-coupling regime of quantum electrodynamics near a photonic bandgap. This regime is characterized by the emergence of spectrally resolvable new polariton states, similar to the well-known vacuum Rabi splitting in cavity quantum electrodynamics. We will give a more quantitative definition of strong coupling in the following discussion.

Our device consists of 14 unit cells, each of which contains two coplanar waveguide (CPW) sections with different lengths and impedances Z (lo = 0.45 mm, hi = 8 mm and Zlo = 28 Ω, Zhi = 125 Ω). These parameters are chosen so that the band edge is within our measurement window (4–10 GHz) and that the bare photonic crystal has a smooth spectrum. The dispersion relation can be calculated using transfer matrices and is given by

where vp is the phase velocity in the waveguide, ωk is the frequency of the allowed Bloch wave and k is the wavevector in the Brillouin zone. Based on the electric field distribution (Bloch wavefunction), we have purposely placed a transmon qubit in the centre of one unit cell in the middle of the device. This makes the qubit optimally coupled to the second photonic band. Consequently, only this band is taken into account, and a quadratic dispersion relation E = ω0 + α(kk0)2 is further assumed. Our device parameters yield ω0/2π = 7.7 GHz. The complete device image is shown in Fig. 1. It is anchored to the base stage (15 mK) of our dilution refrigerator and connected to a typical 50 Ω measurement chain.

The Hamiltonian of the whole system can be written as

where ωq, ωk are the frequencies of the bare qubit and the electromagnetic mode with wavevector k. ak(σ+) and ak(σ) are the mode (qubit) raising and lowering operators. We have ignored other dissipation channels of the qubit and have also performed a rotating wave approximation. In the single-particle spectrum, there exists a polariton state within the bandgap with the eigenenergy ωb given by the root of the equation

We have already assumed that gkg for all wavevectors k, which is valid in our device design. The solution in the bandgap always exists no matter how far the bare qubit frequency ωq is detuned from the band edge ω0. In real space, the photonic part of this polariton state is exponentially localized around the qubit (hence the bound state) with the localization length L given by the penetration depth . The qubit component of this state can be computed to be Pq = 2(ωbω0)/(3ωbωq − 2ω0), therefore this state is mostly qubit-like deep within the bandgap, while it is mostly photon-like close to the band edge.

In an infinite photonic crystal, this bound state can result in permanent light trapping12 in photonic transport. However, in our finite system, the size of which is comparable to L, it is a leaky bound state with a finite spectral linewidth γ. It can be shown21 that γ is proportional to the overlap of this state’s wavefunction with the externally coupled waveguide $\gamma \phantom{\rule{0.2em}{0ex}}\sim {\text{e}}^{\text{-}{d}_{0}/2L}$, where d0 is the physical length of the device. When probed with a weak signal, this state assists photonic transport within the bandgap; hence, we observe a Lorentzian transmission peak centred at ωb. As the bare qubit frequency ωq is tuned closer to the band edge, the bound state has a larger localization length, and thus carries a larger linewidth.

We measure the bound state linewidth γ and exponentially fit the data to the calculated inverse localization length 1/L (Fig. 2b). This yields the effective device length dfit to be 140 ± 12 mm, in agreement with the length of the whole device d0 = 126 mm. To further validate the above theoretical model, we focus on the cases where the bare qubit frequency is completely within the band. In Fig. 2a, we observe that the bound state peak below the band edge persists while the input signal at the bare qubit frequency is completely reflected due to destructive interference15. Now we can extract ωq, ωb and fit the data to equation (3). Note that when the bare qubit is resonant with the band edge, the predicted energy shift is Δ/2π = (ωbω0)/2π = (πg2/α)2/3(1/h) and Pq = 2/3. We use Δ/2π as the fitting parameter instead of g so that we can then define the strong-coupling regime as Δ κ, where κ characterizes the steepness of the band edge. In our device, the best fit yields Δ/2π = 250 MHz, while κ/2π ≈ 26 MHz (see Supplementary Information).

Unlike harmonic defect states, this bound state can be used to control quantum transport within the bandgap. We achieve this by taking advantage of the anharmonic multilevel structure of the transmon qubit (anharmonicity Ec/2π = 385 MHz). We tune the bound state deep into the gap ω0ωb Δ, resonantly pump it with Rabi rate Ωp, and apply another weak tone to probe the transmission. In the limit of Ωp γ, we observe, in Fig. 3a, four extra transmission peaks due to the appearance of Rabi sidebands and an Autler–Townes (AT) splitting of ω12, while the bound state peak is strongly suppressed. The AT splitting arises due to the Rabi splitting of the |1〉 ↔ |2〉 transition. Furthermore, the AT splitting doublet has a much larger transmission amplitude than the Rabi sidebands. Similarly, in Fig. 3b, when the pump tone is resonant with the second transition ωp = ω12, we only observe the AT splitting of ω01, also known as electromagnetically induced transparency (EIT) of a single atom. Here the EIT effect is revealed in the suppression of transmission at ω01 within the bandgap.

These observations are attributed to the coupling of the laser-dressed states with the photonic crystal and their different steady-state populations. Essentially, a photon can transmit through the bandgap only if the resonant dressed-state transition |ν, N〉 ↔ |μ, N + 1〉 is strongly coupled to the waveguide and not population inverted (ρνν > ρμμ), where ρνν = 〈ν |ρ |ν〉 and |ν, N〉 indicate laser-dressed states that include N laser photons. This result can be arrived at by assuming a linear response and using a transfer matrix technique22. In a waveguide with linear dispersion, the resonant transmission coefficient tq of a dressed-state transition can be simplified as23,

where 0 < η ≤ 1 is a quantity that characterizes the coupling between the dressed-state transition and the waveguide. Combining this with transfer matrices of periodic waveguides allows us to determine the total transmission coefficient t in the bandgap. It is readily apparent that the probe signal is amplified (attenuated) when the population is inverted (not inverted) in a normal waveguide, while the opposite is true within the photonic bandgap. For instance, in Fig. 3a near the bound state frequency ω01, when Ωp γ, tq ≈ 0 and rq ≈ −1, yielding t ≈ 1. While for Ωp γ, the corresponding dressed states are almost equally populated, equivalent to tq ≈ 1 and rq ≈ 0, resulting in t ≈ 0. Here rq is the reflection coefficient of the qubit. These calculations yield good agreement, and show quantum transport within the bandgap can indeed be coherently controlled with an external drive (see Supplementary Information).

Laser-dressed states can even hybridize with the photonic crystal and form doubly dressed states, just as a single-photon bound state is formed when a bare qubit is tuned near the band edge. Although being a archetypal quantum optics model, analytical treatment of resonance fluorescence near the band edge is not available24. We present an experimental examination of the driven dynamics as we tune the bound state closer to the band edge. In the pump–probe experiment in Fig. 4a, we observe that when one sideband gets close to the band edge it splits into two resonances, including a peak within the bandgap and a dip within the band. This level splitting (90 MHz) is weaker than the direct coupling between the qubit and the photonic crystal Δ (see Supplementary Information). This spectral information underlies the non-Markovian light emission dynamics; that is, the emitted light at the sideband can be reflected back by the photonic medium and reabsorbed by the qubit.

The deep transmission dip around the upper sideband can be interpreted as dressed-state cooling, which means that the qubit is dynamically pumped into one specific quantum state. Ignoring the band edge effect and higher transmon levels, the reduced dynamics of the qubit can be described by the following master equation (see Supplementary Information)

Here the decay rates of Mollow triplets, γ0, ±, are proportional to the local photonic DOS and are Pauli matrices in the dressed state basis. It is clear that the steady-state population of the dressed state |−〉 is ρ− − = γ+/(γ + γ+). As a result, the qubit will be polarized to the |−〉 state if the upper sideband falls in the photonic band while the lower sideband falls in the bandgap (γ+ γ). Linewidth data in Fig. 2a are used to estimate γ± in the system. Further taking into account the higher transmon level, we then approximate the dressed-state purity as ρ− − ≈ 1 − 2γ/γ+ (method A). Alternatively, the attenuation of the probe signal can be used to deduce ρ− − ≈ 1 − tq/2 based on the above linear response theory (method B). Detailed quantitative analysis using these two methods is given in the Supplementary Information and the results are shown in Fig. 4c. Simulated ρ− − reaches a maximum of 83% near the band edge where γ+ is largest.

In comparison with a prior reservoir engineering approach in which a similar effect was demonstrated25, here only one external drive is required and the cooling effect is caused by the coloured vacuum of the photonic crystal. This mechanism could be used to stabilize an arbitrary state on the Bloch sphere by detuning the drive from the qubit. Future versions of the device can purposely incorporate a defect cavity mode to assist dispersive readout of the qubit state and incorporate more unit cells to increase the cooling fidelity. Furthermore, this can be directly generalized to the many-qubit case where a dynamical quantum phase transition would be observable and highly entangled many-body states can be stabilized26,27. Also, by engineering the coupling of the qubit with multiple bands, a potential wideband dressed-state laser and amplifier can be envisioned.

In the future, low-noise amplifiers can be integrated to study quantum correlation effects in coherent multiphoton transport. The superconducting transmon qubit can be replaced by a flux qubit to reach the ultrastrong-coupling regime28,29. The concept can be generalized to three-dimensional architectures and other quantum emitters. Finally, this provides a platform for studying spin models with coupling mediated by overlapping photonics bound states3,4, with built-in initialization through reservoir engineering.

## Methods

The photonic crystal was made using standard optical lithography and dry etching techniques from a 200 nm Nb thin film on a 10 mm × 10 mm sapphire substrate. The pair of Josephson junctions of the transmon qubit were made using the Dolan bridge technique and evaporated with aluminium. The whole device is packaged in a printed circuit board, wire bonded and anchored at the base plate (15 mK) of our dilution refrigerator. An external solenoid magnet is used to apply a magnetic field across the SQUID loop of the qubit.

### Data availability.

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.