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Letters Published Online: 25 July 2010 | Doi: 10.1038/Nphys1730 LETTERS PUBLISHED ONLINE: 25 JULY 2010 | DOI: 10.1038/NPHYS1730 Circuit quantum electrodynamics in the ultrastrong-coupling regime T. Niemczyk1*, F. Deppe1,2, H. Huebl1, E. P. Menzel1, F. Hocke1, M. J. Schwarz1, J. J. Garcia-Ripoll3, D. Zueco4, T. Hümmer5, E. Solano6,7, A. Marx1 and R. Gross1,2 In circuit quantum electrodynamics1–10 (QED), where super- and a coplanar waveguide resonator. The transmission spectra conducting artificial atoms are coupled to on-chip cavities, the of the combined system reveal qubit–mode couplings g=!r exploration of fundamental quantum physics in the strong- of up to 12% and anticrossings that cannot be explained by coupling regime has greatly evolved. In this regime, an the Jaynes–Cummings model. Instead, they are caused by the atom and a cavity can exchange a photon frequently before simultaneous creation (annihilation) of two excitations, one coherence is lost. Nevertheless, all experiments so far are well in the qubit and one in a resonator mode, while annihilating described by the renowned Jaynes–Cummings model11. Here, (creating) only one excitation in a different resonator mode. we report on the first experimental realization of a circuit QED The size of the anticrossings illustrates the importance of the system operating in the ultrastrong-coupling limit12,13, where counter-rotating terms for the qubit–cavity dynamics in the the atom–cavity coupling rate g reaches a considerable fraction ultrastrong-coupling limit. of the cavity transition frequency !r. Furthermore, we present Images of our quantum circuit and a schematic of the direct evidence for the breakdown of the Jaynes–Cummings measurement set-up are shown in Fig. 1. At a current antinode for model. We reach remarkable normalized coupling rates g=!r of the λ-mode of a niobium superconducting resonator (Fig. 1a–c), a up to 12% by enhancing the inductive coupling14 of a flux qubit part of the centre conductor is replaced with a narrow aluminium to a transmission line resonator. Our circuit extends the toolbox strip interrupted by a large-area Josephson junction (see the of quantum optics on a chip towards exciting explorations of Methods section for fabrication details). This junction mediates ultrastrong light–matter interaction. most of the inductive coupling between a superconducting flux At microwave frequencies, strong coupling is feasible because qubit25 galvanically connected to the strip. The qubit consists of the enormous design flexibility of superconducting circuit QED of three nanometre-scaled Josephson junctions interrupting a systems1,2. Here, small cavity-mode volumes and large dipole superconducting loop, which is threaded by an external flux bias 15 16,17 moments of artificial atoms enable coupling rates g of about Φx . Scanning electron microscope (SEM) images of the qubit loop 1% of the cavity-mode frequency !r. Nevertheless, as in other and the Josephson junctions are shown in Fig. 1d–f. For suitable systems18–22, the quantum dynamics of these strongly coupled junction sizes, the qubit potential landscape can be reduced to systems follows the Jaynes–Cummings model, which describes the a double-well potential, where the two minima correspond to coherent exchange of a single excitation between the atom and states with clockwise and anticlockwise persistent currents Ip. At the cavity mode. Although the Hamiltonian of a realistic atom– δΦx D Φx −Φ0=2 D 0, these two states are degenerate and separated cavity system contains so-called counter-rotating terms allowing by an energy gap ∆. In the qubit eigenbasis,p the qubit Hamiltonian 2 2 the simultaneous creation or annihilation of an excitation in reads Hˆ q D hN!qσˆz =2. Here, !q D ∆ C(2Ip ·δΦx ) =hN is the qubit both, atom and cavity mode, these terms can be safely neglected transition frequency, which can be adjusted by an external flux for small normalized coupling rates g=!r. However, when g bias. We note, that for our flux qubit the two-level approximation becomes a significant fraction of !r, the counter-rotating terms are is well justified because of its large anharmonicity. The resonator ˆ D ! y C = expected to manifest, giving rise to exciting effects in QED. This modes are described as harmonic oscillators, Hn hN n(ˆan ˆan 1 2), ultrastrong-coupling regime is difficult to reach in quantum-optical where !n is the resonance frequency and n is the resonator-mode y cavity QED (refs 19,20), but was recently realized in a solid-state index. The operator ˆan (ˆan) creates (annihilates) a photon in the semiconductor system23,24. There, quantitative deviations from nth resonator mode. Owing to the inhomogeneous transmission the Jaynes–Cummings model have been observed, but direct line geometry14 (see Fig. 1d), the higher mode frequencies of our experimental proof of its breakdown by means of an unambiguous resonator are not integer multiples of the fundamental resonance feature is still missing. frequency !1. Throughout this work, we refer to the nth mode as In this work, we exploit the potential of flux-based the nλ/2-mode. Then, the Hamiltonian of our quantum circuit superconducting quantum circuits to reach the ultrastrong-coupling can be written as regime13,14 and show direct evidence of physics beyond the ˆ D ˆ CX ˆ C y C . θσ − θσ / Jaynes–Cummings model. To this end, we use the large nonlinear H Hq Hn hgN n ˆan ˆan cos ˆz sin ˆx (1) inductance of a Josephson junction shared between a flux qubit n 1Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, D-85748 Garching, Germany, 2Physik-Department, Technische Universität München, D-85748 Garching, Germany, 3Instituto de Física Fundamental, CSIC, Serrano 113-bis, 28006 Madrid, Spain, 4Instituto de Ciencia de Materiales de Aragón and Departamento de Física de la Materia Condensada, CSIC-Universidad de Zaragoza, E-50009 Zaragoza, Spain, 5Institut für Physik, Universität Augsburg, Universitätsstraße 1, D-86135 Augsburg, Germany, 6Departamento de Química Física, Universidad del País Vasco - Euskal Herriko Unibertsitatea, Apdo. 644, 48080 Bilbao, Spain, 7IKERBASQUE, Basque Foundation for Science, Alameda Urquijo 36, 48011 Bilbao, Spain. *e-mail: [email protected]. 772 NATURE PHYSICS j VOL 6 j OCTOBER 2010 j www.nature.com/naturephysics © 2010 Macmillan Publishers Limited. All rights reserved. NATURE PHYSICS DOI: 10.1038/NPHYS1730 LETTERS a bcI (x) 0.1 mm x 0 Qubit L position 5 mm d g μ ω ω 20 m s VNA rf 300 K Amplifier 4 K Cryogenic amplifier 0.6 K Circulator Attenuated input line Qubit 15 mK ef Resonator 1 μm 1 μm Cκ LJ Cκ Figure 1 j Quantum circuit and experimental set-up. a, Optical image of the superconducting λ/2 coplanar waveguide resonator (light blue rectangle). Black rectangles: area shown in b. Red rectangle: area shown in d. b, SEM image of one of the coupling capacitors. c, Sketch of the current distribution of the first three resonator modes. Their resonance frequencies are !1=2π D 2:782 GHz (λ/2, red), !2=2π D 5:357 GHz (λ, blue) and !3=2π D 7:777 GHz (3λ/2, green). The cavity modes !n are measured at maximum qubit–cavity detuning (Φx D 0). In general, the flux dependence of !n is very weak, except for the regions close to Φx D Φ0=2. d, SEM image of the galvanically coupled flux qubit (see the Methods section for fabrication details). The width in the overlap regions with the centre conductor is 20 µm, and that of the constriction is 1 µm. Orange rectangle: area shown in e. Green rectangle: area shown in f. e, SEM image of the large Josephson junction. Its Josephson inductance LJ is responsible for approximately 85% of the qubit–resonator coupling. f, One Josephson junction of the qubit loop. The area of this junction is 14% of the one shown in e. g, Schematic of the measurement set-up. The transmission through the cavity at !rf is measured using a vector network analyser (VNA). A second microwave signal at !s is used for two-tone qubit spectroscopy. The input signal is attenuated at various temperature stages and coupled into the resonator (light blue) through the capacitors Cκ . The crossed squares represent Josephson junctions. A circulator isolates the sample from the amplifier noise. Here, σˆx;z denote Pauli operators, gn is the coupling rate of the gap ∆ and the coupling constants gn in this situation. Instead, we qubit to the nth cavity mode and the flux dependence is encoded extract them from a cavity transmission spectrum with negligible in sin θ D ∆=hN!q and cos θ. The operator σˆx is conveniently photon population. For that purpose, we first measure the power- expressed as the sum of the qubit raising (σˆC) and lowering (σˆ−) dependent a.c.-Zeeman shift of the qubit transition frequency operator. Thus, in contrast to the Jaynes–Cummings model, the at δΦx D 0. The data are shown in the inset of Fig. 2a. The Hamiltonian in equation (1) explicitly contains counter-rotating average photon number ¯n3 can be estimated using the relation yσ σ D ! κ κ = ≈ : terms of the form ˆan ˆC and ˆan ˆ−. Figure 1g shows a schematic of Prf ¯n3hN 3 3 (refs 6,8), where 3 2π 3 7 MHz is the full-width our measurement set-up. The quantum circuit is located at the base at half-maximum of the cavity resonance and Prf is the probe power temperature of 15 mK in a dilution refrigerator. We measure the referred to the input of the resonator. Figure 2b shows a colour- amplified resonator transmission using a vector network analyser. coded transmission spectrum for the 3λ/2-mode as a function of For qubit spectroscopy measurements, the system is excited with δΦx .
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