Abstract
Photonic crystals are a powerful tool for the manipulation of optical dispersion and density of states, and have thus been used in applications from photon generation to quantum sensing with nitrogen vacancy centres and atoms^{1,2}. The unique control provided by these media makes them a beautiful, if unexplored, playground for strongcoupling quantum electrodynamics, where a single, highly nonlinear emitter hybridizes with the band structure of the crystal. Here we demonstrate that such a hybridization can create localized cavity modes that live within the photonic bandgap, whose localization and spectral properties we explore in detail. We then demonstrate that the coloured vacuum of the photonic crystal can be employed for efficient dissipative state preparation. This work opens exciting prospects for engineering longrange spin models^{3,4} in the circuit quantum electrodynamics architecture, as well as new opportunities for dissipative quantum state engineering.
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 dots^{5,6}, magnons^{7} and superconducting qubits^{8}. However, when an atom is strongly coupled to a photonic crystal, nonperturbative effects become important, and significantly enrich the physics. For instance, a singlephoton bound state has been predicted to emerge within the gap^{9}, and spontaneous emission of the atom will thus exhibit Rabi oscillation and lighttrapping 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 nature^{10,11,12}. In contrast to a system with discrete cavity modes, which is well described by the singlemode or multimode Jaynes–Cummings Hamiltonian^{13,14}, a continuous density of states enables the formation of a localized state in the bandgap. While other spinboson problems with continuous DOS have also been studied experimentally^{15,16} or theoretically^{17,18} 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 crystals^{19,20}, 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 waveguides^{20}, 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 strongcoupling regime of quantum electrodynamics near a photonic bandgap. This regime is characterized by the emergence of spectrally resolvable new polariton states, similar to the wellknown 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 ℓ and impedances Z (ℓ_{lo} = 0.45 mm, ℓ_{hi} = 8 mm and Z_{lo} = 28 Ω, Z_{hi} = 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 v_{p} is the phase velocity in the waveguide, ω_{k} 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 = ℏω_{0} + α(k − k_{0})^{2} is further assumed. Our device parameters yield ω_{0}/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 ω_{q}, ω_{k} are the frequencies of the bare qubit and the electromagnetic mode with wavevector k. a_{k}^{†}(σ^{+}) and a_{k}(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 singleparticle spectrum, there exists a polariton state within the bandgap with the eigenenergy ω_{b} given by the root of the equation
We have already assumed that g_{k} ≈ g 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 ω_{q} is detuned from the band edge ω_{0}. 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 P_{q} = 2(ω_{b} − ω_{0})/(3ω_{b} − ω_{q} − 2ω_{0}), therefore this state is mostly qubitlike deep within the bandgap, while it is mostly photonlike close to the band edge.
In an infinite photonic crystal, this bound state can result in permanent light trapping^{12} 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 shown^{21} 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 d_{0} 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 ω_{b}. As the bare qubit frequency ω_{q} 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 d_{fit} to be 140 ± 12 mm, in agreement with the length of the whole device d_{0} = 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 interference^{15}. Now we can extract ω_{q}, ω_{b} 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 = (ω_{b} − ω_{0})/2π = (πg^{2}/α)^{2/3}(1/h) and P_{q} = 2/3. We use Δ/2π as the fitting parameter instead of g so that we can then define the strongcoupling 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 E_{c}/2π = 385 MHz). We tune the bound state deep into the gap ω_{0} − ω_{b} ≫ Δ, resonantly pump it with Rabi rate Ω_{p}, and apply another weak tone to probe the transmission. In the limit of Ω_{p} ≫ γ, we observe, in Fig. 3a, four extra transmission peaks due to the appearance of Rabi sidebands and an Autler–Townes (AT) splitting of ω_{12}, 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 ω_{p} = ω_{12}, we only observe the AT splitting of ω_{01}, also known as electromagnetically induced transparency (EIT) of a single atom. Here the EIT effect is revealed in the suppression of transmission at ω_{01} within the bandgap.
These observations are attributed to the coupling of the laserdressed states with the photonic crystal and their different steadystate populations. Essentially, a photon can transmit through the bandgap only if the resonant dressedstate transition ν, N〉 ↔ μ, N + 1〉 is strongly coupled to the waveguide and not population inverted (ρ_{νν} > ρ_{μμ}), where ρ_{νν} = 〈ν ρ ν〉 and ν, N〉 indicate laserdressed states that include N laser photons. This result can be arrived at by assuming a linear response and using a transfer matrix technique^{22}. In a waveguide with linear dispersion, the resonant transmission coefficient t_{q} of a dressedstate transition can be simplified as^{23},
where 0 < η ≤ 1 is a quantity that characterizes the coupling between the dressedstate 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 ω_{01}, when Ω_{p} ≪ γ, t_{q} ≈ 0 and r_{q} ≈ −1, yielding t ≈ 1. While for Ω_{p} ≫ γ, the corresponding dressed states are almost equally populated, equivalent to t_{q} ≈ 1 and r_{q} ≈ 0, resulting in t ≈ 0. Here r_{q} 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).
Laserdressed states can even hybridize with the photonic crystal and form doubly dressed states, just as a singlephoton 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 available^{24}. 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 nonMarkovian 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 dressedstate 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, γ_{0, ±}, are proportional to the local photonic DOS and are Pauli matrices in the dressed state basis. It is clear that the steadystate 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 dressedstate purity as ρ_{− −} ≈ 1 − 2γ_{−}/γ_{+} (method A). Alternatively, the attenuation of the probe signal can be used to deduce ρ_{− −} ≈ 1 − t_{q}/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 demonstrated^{25}, 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 manyqubit case where a dynamical quantum phase transition would be observable and highly entangled manybody states can be stabilized^{26,27}. Also, by engineering the coupling of the qubit with multiple bands, a potential wideband dressedstate laser and amplifier can be envisioned.
In the future, lownoise 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 ultrastrongcoupling regime^{28,29}. The concept can be generalized to threedimensional architectures and other quantum emitters. Finally, this provides a platform for studying spin models with coupling mediated by overlapping photonics bound states^{3,4}, with builtin 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.
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Acknowledgements
The authors would like to acknowledge D. Sadri, G. Zhang, N. M. Sundaresan and J. Simon for valuable discussions. This work is supported by IARPA under contract W911NF1010324 and the US National Science Foundation through Materials Research Science and Engineering Centers under contract DMR1420541.
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Y.L. designed the device, performed the measurements and analysed the data. A.A.H. supervised the whole experiment. All authors contributed to the preparation of this manuscript.
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Liu, Y., Houck, A. Quantum electrodynamics near a photonic bandgap. Nature Phys 13, 48–52 (2017). https://doi.org/10.1038/nphys3834
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DOI: https://doi.org/10.1038/nphys3834
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