ARTICLE DOI: 10.1038/s41467-018-06142-z OPEN Superconducting metamaterials for waveguide quantum electrodynamics Mohammad Mirhosseini1,2,3, Eunjong Kim1,2,3, Vinicius S. Ferreira1,2,3, Mahmoud Kalaee1,2,3, Alp Sipahigil 1,2,3, Andrew J. Keller1,2,3 & Oskar Painter1,2,3 Embedding tunable quantum emitters in a photonic bandgap structure enables control of dissipative and dispersive interactions between emitters and their photonic bath. Operation in 1234567890():,; the transmission band, outside the gap, allows for studying waveguide quantum electro- dynamics in the slow-light regime. Alternatively, tuning the emitter into the bandgap results in finite-range emitter–emitter interactions via bound photonic states. Here, we couple a transmon qubit to a superconducting metamaterial with a deep sub-wavelength lattice constant (λ/60). The metamaterial is formed by periodically loading a transmission line with compact, low-loss, low-disorder lumped-element microwave resonators. Tuning the qubit frequency in the vicinity of a band-edge with a group index of ng = 450, we observe an anomalous Lamb shift of −28 MHz accompanied by a 24-fold enhancement in the qubit lifetime. In addition, we demonstrate selective enhancement and inhibition of spontaneous emission of different transmon transitions, which provide simultaneous access to short-lived radiatively damped and long-lived metastable qubit states. 1 Kavli Nanoscience Institute, California Institute of Technology, Pasadena, CA 91125, USA. 2 Thomas J. Watson, Sr., Laboratory of Applied Physics, California Institute of Technology, Pasadena, CA 91125, USA. 3 Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA. Correspondence and requests for materials should be addressed to O.P. (email: [email protected]) NATURE COMMUNICATIONS | (2018) 9:3706 | DOI: 10.1038/s41467-018-06142-z | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-06142-z avity quantum electrodynamics (QED) studies the inter- In this paper, we utilize an array of coupled lumped-element action of an atom with a single electromagnetic mode of a microwave resonators to form a compact bandgap waveguide C fi 1,2 λ high- nesse cavity with a discrete spectrum . In this with a deep sub-wavelength lattice constant ( /60) based on the canonical setting, a large photon–atom coupling is achieved by metamaterial concept. In addition to a compact footprint, these repeated interaction of the atom with a single photon bouncing sort of structures can exhibit highly nonlinear band dispersion many times between the cavity mirrors. Recently, there has been surrounding the bandgap, leading to exceptionally strong con- much interest in achieving strong light–matter interaction in a finement of localized intra-gap photon states. We present the cavity-free system such as a waveguide3,4. Waveguide QED refers design and fabrication of such a metamaterial waveguide, and to a system where a chain of atoms are coupled to a common characterize the resulting waveguide dispersion and bandgap optical channel with a continuum of electromagnetic modes over properties via interaction with a tunable superconducting trans- a large bandwidth. Slow-light photonic crystal waveguides are of mon qubit. We measure the Lamb shift and lifetime of the qubit particular interest in waveguide QED because the reduced group in the bandgap and its vicinity, demonstrating the anomalous velocity near a bandgap preferentially amplifies the desired Lamb shift of the fundamental qubit transition as well as selective radiation of the atoms into the waveguide modes5–7. Moreover, in inhibition and enhancement of spontaneous emission for the first this configuration an interesting paradigm can be achieved by two excited states of the transmon qubit. placing the resonance frequency of the atom inside the bandgap – of the waveguide8 11. In this case, the atom cannot radiate into Results the waveguide but the evanescent field surrounding it gives rise to Band-structure analysis and spectroscopy. We begin by con- a photonic bound state9. The interaction of such localized bound sidering the circuit model of a CPW that is periodically loaded states has been proposed for realizing tunable spin–exchange with microwave resonators as shown in the inset to Fig. 1a. The interaction between atoms in a chain12,13, and also for realizing Lagrangian for this system can be constructed as a function of the effective non-local interactions between photons14,15. fl Φb Φa37 node uxes of the resonator and waveguide sections n and n . While achieving efficient waveguide coupling in the optical Assuming periodic boundary conditions and applying the rotat- regime requires the challenging task of interfacing atoms or ing wave approximation, we derive the Hamiltonian for this 16–20 ω atomic-like systems with nanoscale dielectric structures , system and solve for the energies h ± ;k along with the corre- ji¼; α^ ji superconducting circuits provide an entirely different platform sponding eigenstates ± k ± ;k 0 as (see Supplementary for studying the physics of light–matter interaction in the Note 1) microwave regime4,21. Development of the field of circuit QED qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi has enabled fabrication of tunable qubits with long coherence 1 2 2 ω ; ¼ ðÞΩ þ ω ± ðÞΩ À ω þ4g ; ð1Þ times and fast qubit gate times22,23. Moreover, strong coupling is ± k 2 k 0 k 0 k readily achieved in coplanar platforms due to the deep sub- wavelength transverse confinement of photons attainable in ω À ω microwave waveguides and the large electric dipole of super- ± ;k 0 24 α^ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigk ^ : conducting qubits . Microwave waveguides with strong disper- ± ;k ak bk ð Þ 2 2 2 sion, even “bandgaps” in frequency, can also be simply realized by 2 2 ω ; À ω þg ω ; À ω þg periodically modulating the geometry of a coplanar transmission ± k 0 k ± k 0 k line25. Such an approach was recently demonstrated in a pio- neering experiment by Liu and Houck26, whereby a qubit was ^ ^ coupled to the localized photonic state within the bandgap of a Here, ak and bk describe the momentum-space annihilation modulated coplanar waveguide (CPW). Satisfying the Bragg operators for the bare waveguide and bare resonator sections, the Ω ω condition in a periodically modulated waveguide requires a lattice index k denotes the wavevector, and the parameters k, 0, and gk constant on the order of the wavelength, however, which trans- quantify the frequency of traveling modes of the bare waveguide, lates to a device size of approximately a few centimeters for the resonance frequency of the microwave resonators, and complete confinement of the evanescent fields in the frequency coupling rate between resonator and waveguide modes, respec- α^ range suitable for microwave qubits. Such a restriction sig- tively. The operators ± ;k represent quasi-particle solutions of the nificantly limits the scaling in this approach, both in qubit composite waveguide, where far from the bandgap the quasi- number and qubit connectivity. particle is primarily composed of the bare waveguide mode, while ω fi An alternative approach for tailoring dispersion in the in the vicinity of 0 most of its energy is con ned in the microwave domain is to take advantage of the metamaterial microwave resonators. ω concept. Metamaterials are composite structures with sub- Figure 1a depicts the numerically calculated energy bands ±,k wavelength components, which are designed to provide an as a function of the wavevector k. It is evident that the dispersion effective electromagnetic response27,28. Since the early micro- has the form of an avoided crossing between the energy bands of wave work, the electromagnetic metamaterial concept has been the bare waveguide and the uncoupled resonators. For small gap expanded and extensively studied across a broad range of sizes, the midgap frequency is close to the resonance frequency of 29–31 ω classical optical sciences ; however, their role in quantum uncoupled resonators 0, and unlike the case of a periodically optics has remained relatively unexplored, at least in part due to modulated waveguide, there is no fundamental relation tying the the lossy nature of many sub-wavelength components. midgap frequency to the lattice constant in this case. The form of ω Improvements in design and fabrication of low-loss super- the band structure near the higher cut-off frequency c+ can be ω − ω ∝ 2 conducting circuit components in circuit QED offer a new approximated as a quadratic function ( c+) k , whereas the ω prospect for utilizing microwave metamaterials in quantum band structure near the lower band-edge c− is inversely 32 ω − ω ∝ 2 applications . Indeed, high quality-factor superconducting proportional to the square of the wavenumber ( c−) 1/k . components such as resonators can be readily fabricated on a The analysis above has been presented for resonators which are chip33, and such elements have been used as a tool for achieving capacitively coupled to a waveguide in a parallel geometry; a phase-matching in near quantum-limited traveling wave similar band structure can also be achieved using series inductive amplifiers34,35 and for tailoring qubit interactions in a multi- coupling of resonators (see Supplementary Note 1 and Supple- mode cavity QED architecture36. mentary Fig. 1). 2 NATURE COMMUNICATIONS | (2018) 9:3706 | DOI: 10.1038/s41467-018-06142-z | www.nature.com/naturecommunications NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-06142-z ARTICLE abthe resonators be of high impedance. Use of high impedance 10 x resonators allows for a larger photonic bandgap and greater +, k waveguide–qubit coupling. For the waveguide QED application of 9 interest this enables denser qubit circuits, both spatially and Bare waveguide 1/2 Ck 8 dispersion spectrally. The impedance of the resonators scales roughly as the inverse 7 square-root of the pitch of the wires in the spiral coils. 0/2 6 Bandgap region Complicating matters is that smaller wire widths have been (imaginary k-vector) Symmetry axis found to introduce larger resonator frequency disorder due to 5 L 39 r kinetic inductance effects .