Circuit Quantum Electrodynamics
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REVIEWS OF MODERN PHYSICS, VOLUME 93, APRIL–JUNE 2021 Circuit quantum electrodynamics Alexandre Blais Institut quantique and D´epartement de Physique, Universit´e de Sherbrooke, Sherbrooke, Qu´ebec J1K 2R1, Canada and Canadian Institute for Advanced Research, Toronto M5G 1M1, Ontario, Canada Arne L. Grimsmo Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, New South Wales 2006, Australia S. M. Girvin Yale Quantum Institute, P.O. Box 208 334, New Haven, Connecticut 06520-8263, USA Andreas Wallraff Department of Physics, ETH Zurich, CH-8093 Zurich, Switzerland and Quantum Center, ETH Zurich, CH-8093 Zurich, Switzerland (published 19 May 2021) Quantum-mechanical effects at the macroscopic level were first explored in Josephson-junction- based superconducting circuits in the 1980s. In recent decades, the emergence of quantum information science has intensified research toward using these circuits as qubits in quantum information processors. The realization that superconducting qubits can be made to strongly and controllably interact with microwave photons, the quantized electromagnetic fields stored in superconducting circuits, led to the creation of the field of circuit quantum electrodynamics (QED), the topic of this review. While atomic cavity QED inspired many of the early developments of circuit QED, the latter has now become an independent and thriving field of research in its own right. Circuit QED allows the study and control of light-matter interaction at the quantum level in unprecedented detail. It also plays an essential role in all current approaches to gate-based digital quantum information processing with superconducting circuits. In addition, circuit QED provides a framework for the study of hybrid quantum systems, such as quantum dots, magnons, Rydberg atoms, surface acoustic waves, and mechanical systems interacting with microwave photons. Here the coherent coupling of superconducting qubits to microwave photons in high-quality oscillators focusing on the physics of the Jaynes-Cummings model, its dispersive limit, and the different regimes of light-matter interaction in this system are reviewed. Also discussed is coupling of superconducting circuits to their environment, which is necessary for coherent control and measurements in circuit QED, but which also invariably leads to decoherence. Dispersive qubit readout, a central ingredient in almost all circuit QED experiments, is also described. Following an introduction to these fundamental concepts that are at the heart of circuit QED, important use cases of these ideas in quantum information processing and in quantum optics are discussed. Circuit QED realizes a broad set of concepts that open up new possibilities for the study of quantum physics at the macro scale with superconducting circuits and applications to quantum information science in the widest sense. DOI: 10.1103/RevModPhys.93.025005 CONTENTS A. Exchange interaction between a transmon and an oscillator 12 I. Introduction 2 B. The Jaynes-Cummings spectrum 13 II. Superconducting Quantum Circuits 3 C. Dispersive regime 14 A. The quantum LC resonator 3 1. Schrieffer-Wolff approach 14 B. 2D resonators 4 2. Bogoliubov approach 15 1. Quantized modes of the transmission-line D. Josephson junctions embedded in multimode resonator 6 electromagnetic environments 16 C. 3D resonators 7 E. Beyond the transmon: Multilevel artificial atom 17 D. The transmon artificial atom 8 F. Alternative coupling schemes 18 E. Flux-tunable transmons 11 IV. Coupling to the Environment 18 F. Other superconducting qubits 11 A. Wiring up quantum systems with transmission lines 18 III. Light-Matter Interaction in Circuit QED 12 B. Input-output theory in electrical networks 19 0034-6861=2021=93(2)=025005(72) 025005-1 © 2021 American Physical Society Blais, Grimsmo, Girvin, and Wallraff: Circuit quantum electrodynamics C. Qubit relaxation and dephasing 20 I. INTRODUCTION D. Dissipation in the dispersive regime 21 E. Multimode Purcell effect and Purcell filters 22 Circuit quantum electrodynamics (QED) is the study of the F. Controlling quantum systems with microwave drives 23 interaction of nonlinear superconducting circuits, acting as V. Measurements in Circuit QED 23 artificial atoms or as qubits for quantum information process- A. Microwave field detection 23 ing, with quantized electromagnetic fields in the microwave- B. Phase-space representations and their relation to field frequency domain. Inspired by cavity QED (Kimble, 1998; detection 26 Haroche and Raimond, 2006), a field of research originating C. Dispersive qubit readout 27 from atomic physics and quantum optics, circuit QED has led 1. Steady-state intracavity field 27 to advances in the fundamental study of light-matter inter- 2. Signal-to-noise ratio and measurement fidelity 29 action, in the development of quantum information processing 3. Other approaches to qubit readout 31 technology (Clarke and Wilhelm, 2008; Wendin, 2017; Krantz a. Josephson bifurcation amplifier 31 et al., 2019; Blais, Girvin, and Oliver, 2020; Kjaergaard, b. High-power readout and qubit “punch out” 31 c. Squeezing 31 Schwartz et al., 2020), and in the exploration of novel hybrid d. Longitudinal readout 31 quantum systems (Xiang et al., 2013; Clerk et al., 2020). VI. Qubit-Resonator Coupling Regimes 32 First steps toward exploring the quantum physics of super- A. Resonant regime 32 conducting circuits were made in the mid 1980s. At that time, 1. Bad-cavity limit 32 the question arose as to whether quantum phenomena, such as 2. Bad-qubit limit 33 quantum tunneling or energy level quantization, could be 3. Strong-coupling regime 33 observed in macroscopic systems of any kind (Leggett, 1980, B. Dispersive regime 35 1984b). One example of such a macroscopic system is the 1. Qubit spectroscopy 36 Josephson tunnel junction (Josephson, 1962; Tinkham, 2004) 2. ac-Stark shift and measurement-induced formed by a thin insulating barrier at the interface between broadening 36 two superconductors and in which macroscopic quantities C. Beyond strong coupling: Ultrastrong-coupling regime 38 such as the current flowing through the junction or the voltage VII. Quantum Computing With Circuit QED 39 developed across it are governed by the dynamics of the A. Single-qubit gates 39 gauge-invariant phase difference of the Cooper pair conden- B. Two-qubit gates 41 sate across the junction. The first experimental evidence for 1. Qubit-qubit exchange interaction 41 quantum effects in these circuits (Clarke et al., 1988) was the a. Direct capacitive coupling 41 observation of quantum tunneling of the phase degree of b. Resonator mediated coupling 42 freedom of a Josephson junction (Devoret, Martinis, and 2. Flux-tuned 11-02 phase gate 43 Clarke, 1985), rapidly followed by the measurement of 3. All-microwave gates 44 quantized energy levels of the same degree of freedom a. Resonator-induced phase gate 44 (Martinis, Devoret, and Clarke, 1985). b. Cross-resonance gate 45 While the possibility of observation of coherent quantum 4. Parametric gates 46 phenomena in Josephson-junction-based circuits, such as C. Encoding a qubit in an oscillator 47 coherent oscillations between two quantum states of the VIII. Quantum Optics on a Chip 49 junction and the preparation of quantum superpositions, A. Intracavity fields 49 was already envisaged in the 1980s (Tesche, 1987), the B. Quantum-limited amplification 50 prospect of realizing superconducting qubits for quantum C. Propagating fields and their characterization 51 computation revived interest in the pursuit of this goal (Bocko, 1. Itinerant single and multiphoton states 51 Herr, and Feldman, 1997; Shnirman, Schön, and Hermon, 2. Squeezed microwave fields 52 1997; Bouchiat et al., 1998; Makhlin, Schön, and Shnirman, D. Remote entanglement generation 53 1999, 2001). In a groundbreaking experiment, time-resolved E. Waveguide QED 54 coherent oscillations with a superconducting qubit were F. Single-microwave photon detection 55 IX. Outlook 56 observed in 1999 (Nakamura, Pashkin, and Tsai, 1999). Acknowledgments 57 Further progress resulted in the observation of coherent Appendix A: Hamiltonian of a Voltage Biased Transmon 57 oscillations in coupled superconducting qubits (Pashkin et al., 1. Classical gate voltage 58 2003; Yamamoto et al., 2003) and in significant improve- 2. Coupling to an LC oscillator 58 ments of the coherence times of these devices by exploiting Appendix B: Unitary Transformations 59 symmetries in the Hamiltonian underlying the description of 1. Schrieffer-Wolff perturbation theory 59 the circuits (Vion et al., 2002; Clarke and Wilhelm, 2008; 2. Schrieffer-Wolff transformation for a multilevel system Kjaergaard, Schwartz et al., 2020). coupled to an oscillator in the dispersive regime 60 In parallel to these advances, in atomic physics and a. The transmon 60 quantum optics cavity QED developed into an excellent b. The Jaynes-Cummings model 61 setting for the study of the coherent interactions between 3. Bogoliubov approach to the dispersive regime 61 individual atoms and quantum radiation fields (Rempe, 4. Off-resonantly driven transmon 62 Walther, and Klein, 1987; Haroche and Kleppner, 1989; Appendix C: Input-Output Theory 63 Thompson, Rempe, and Kimble, 1992; Brune et al., 1996), References 64 and its application to quantum communication (Kimble, 2008) Rev. Mod. Phys., Vol. 93, No. 2, April–June 2021 025005-2 Blais, Grimsmo, Girvin, and Wallraff: Circuit quantum electrodynamics and quantum computation (Kimble, 1998; Haroche