Nonlinearity Engineering with the Quarton by Yufeng Ye B.ASc., University of Toronto (2019) Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Master of Science in Electrical Engineering and Computer Science at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY May 2020 ○c Massachusetts Institute of Technology 2020. All rights reserved. Author................................................................ Department of Electrical Engineering and Computer Science May 15, 2020 Certified by. Kevin P. O’Brien Assistant Professor of Electrical Engineering and Computer Science Thesis Supervisor Accepted by . Leslie A. Kolodziejski Professor of Electrical Engineering and Computer Science Chair, Department Committee on Graduate Students 2 Nonlinearity Engineering with the Quarton by Yufeng Ye Submitted to the Department of Electrical Engineering and Computer Science on May 15, 2020, in partial fulfillment of the requirements for the degree of Master of Science in Electrical Engineering and Computer Science Abstract In this thesis, we show the principles and applications of a new technique we call “nonlinearity engineering” using a recent superconducting qubit, the Quarton. In tra- ditional nonlinear optics, nonlinear effects are usually weak perturbations to linear interactions. Similarly, microwave quantum optics with superconducting circuits re- lies on the Josephson junction for a negative Kerr nonlinearity that is much weaker than its associated linear energy. Recently, a new superconducting qubit known as the “Quarton” can offer non-perturbatively strong nonlinearity. Here, we demonstrate the general principle of using the Quarton’s positive Kerr and zero linear energy to perform nonlinearity engineering, i.e. the selective design of the nonlinear properties of microwave artifical atoms, metamaterials, and photons in a manner that (tothe best of our knowledge) has no optical analog. We show that for Quarton mediated light-matter coupling, the Quarton can erase or amplify the nonlinearity of artificial atoms and metamaterials. Without nonlinearity, matter behaves light-like and we find (to our best knowledge) the first theoretical demonstration of cross-Kerr between lin- ear microwave photon modes. We extend these fundamental results and provide a practical application by designing a Josephson traveling wave photon detector. Thesis Supervisor: Kevin P. O’Brien Title: Assistant Professor of Electrical Engineering and Computer Science 3 4 Acknowledgments I have been extremely blessed to have met all the amazing people in my life that made this thesis possible. They are my family and friends, my mentors, my peers, and the many kind strangers along the way. This thesis is dedicated to them. I’d like to begin by thanking my advisor Kevin O’Brien. Before joining his group, my mental stereotype of a PhD advisor was that of an old-fashioned scholar per- petually out-of-reach to students, communicates in ancient cryptic languages, and performs black arts in the laboratory. I never imagined having an advisor as open- minded, supportive, and genuinely caring as Kevin. I am extremely grateful for his guidance along the way, both academic and personal. I’d like to thank QCE, Greg, Mahdi, Kaidong, and our collaborators Arne Grimsmo and Will Oliver for their support, helpful insights, and discussions throughout this intellectual journey. I owe a huge debt of gratitude to my previous advisors Alexandre Blais and Nazir Kherani, and mentors Andrew Flood, Baptiste Royer, and Joel Loh. I am thankful for their guidance during my undergrad. Finally, none of this would have been possible without the loving support of my family. I’d like to thank my parents for their unconditional support. I’d like to thank Run for her love and companionship. I’d also like to thank my friends Sophia, Zhen, Yang, Junhong and Guang, Zhuquan, Peter and other friends at the H & M Cooking Group; as well as Ruicong, Yujie, Peiqi, Devany and others from the Avalon game group; and lastly all the amazing people at MIT Singing for Service. My old friends Raymond and Weiran have been with me every step along the way, and I’m forever grateful for their friendship. This work is supported by funding from the MIT Center for Quantum Engineering via support from the Laboratory for Physical Sciences and the MIT EECS department in the form of the Jin Au Kong fellowship. 5 6 Contents 1 Introduction 13 1.1 Thesis structure . 14 2 Background 17 2.1 Superconductivity and the Josephson junction . 17 2.1.1 External flux and gauge choice . 19 2.2 Circuit quantization . 21 2.3 Qubits - artificial atoms . 23 2.4 Waveguides - metamaterials . 25 3 Deriving the Quarton 31 3.1 Derivation . 31 3.1.1 Generality of derivation and unfavorable regimes . 34 3.1.2 Nonlinear optics language: 휒(1); 휒(3) of qubits . 35 3.2 Note on junction capacitance . 36 4 The Quarton as a nonlinear coupler 39 4.1 Literature on circuit QED light-matter coupling . 39 4.2 Erasing and enhancing nonlinearity . 41 4.3 Evaluating Quarton coupling . 43 5 Quarton for traveling wave single photon detection 45 5.1 Literature review on photon detection . 45 5.2 Quarton for JTWPD . 47 7 5.2.1 Multimodes: the Quarton coupled waveguides . 47 5.2.2 Metamaterial design . 49 6 Conclusion 53 6.1 Summary . 53 A Derivation of Quarton enhanced longitudinal qubit coupling gzz 55 A.1 Literature - coupling with linear L ................... 55 A.2 Quarton coupler - coupling without linear L .............. 57 A.2.1 Fundamental gzz ≫ EC limit . 58 8 List of Figures 2-1 a) Superconducting loop threaded by external flux Φe (Φext in text). Panels b) and c) show two equivalent choices of spanning trees (solid lines) and closure branches (dashed lines) [47]. 20 2-2 Two representations of a LC oscillator. a) In terms of “position” vari- able charge q and “momentum” variable flux Φ. b) In terms of “posi- tion” variable flux 휑 and “momentum” variable charge Q. [14] . 21 2-3 Linear resonator vs nonlinear qubit. a) Circuit of linear LC oscillator. b) Energy diagram of the linear resonator. c) Circuit of nonlinear resonator, the cross symbol denotes the JJ. d) Energy diagram of the nonlinear resonator (transmon) that serves as a qubit or artificial atom [25]..................................... 24 2-4 Generic two-port network [35] . 26 2-5 a) Right-handed metamaterial. b) Left-handed metamaterial. c) Com- posite right-/left-handed (CRLH) metamaterial . 28 2-6 Band diagram of CRLH metamaterial showing both left- and right- handed behavior [6]. 28 9 3-1 Quarton as a purely nonlinear element. (a) Schematic plot of the non- 1 linear K vs linear 2L landscape of inductive superconducting elements with centrosymmetric potentials. The Quarton forms a basis for this space along with the inductor. (b) Schematic line scale of the rel- ative strength of nonlinear K of the elements in (a). The Quarton (represented by the spider symbol) defines the infinity limit of this scale. Corresponding potential landscapes are plotted below different elements to visually display the degree of anharmonicity / nonlinearity. 32 4-1 Nonlinearity engineering of light-matter interactions. (a) Circuit of Quarton nonlinearly coupling two qubits. (b) An exact spring-mass analog of (a). (c) Quarton enhancing nonlinearity of qubits, facilitates longitudinal qubit-qubit coupling. (d) Quarton eliminating nonlinear- ity of one qubit, facilitates effective AC-Stark like qubit-photon cou- pling. (e) Quarton eliminating nonlinearity of both qubits, facilitates effective cross-Kerr photon-photon coupling. a;^ b represent annihilation operators for photon modes, 휎z represents qubit modes. 41 4-2 Quarton enabled coupling. (a) Strong longitudinal coupling of trans- mons showing gzz ≫ EC . (b) Cross-Kerr coupling gab of resonators without self-Kerr. 43 5-1 Schematic of a Quarton based JTWPD. Both probe and signal waveg- uides are linear (no self-Kerr) but there is nonlinear coupling (cross- Kerr) between them. 47 5-2 JTWPD circuit consisting of Quarton coupled of left-handed metama- terials with higher frequency cut-offs. Red box outlines the cross-Kerr without self-Kerr circuit shown in Fig. 4-1a, here the JJ acts as an effective inductor. 48 10 5-3 Linear JTWPD metamaterial properties calculated from ABCD ma- trices. a) Band structure b) Characteristic impedance Z0 (grey region highlights impedance matched to 50 ± 2Ω). Clear disjoint passpands and Z0 ! 0 cut-off behavior shown. (’+’ and ’-’ represent right and left moving eigensolutions.) . 50 A-1 C-shunt SQUID (green) nonlinearly coupling two transmons (grey) [31] 56 A-2 JRM facilitated longitudinal qubit coupling gzz [27]. 57 11 12 Chapter 1 Introduction Superconducting quantum circuits has emerged as one of the most promising hardware platforms for quantum information processing [4, 10]. For its high coherence, ease of fabrication, and vast range of engineering tunability, superconducting qubits and metamaterials have been used extensively for many quantum information applications [16]. In quantum computing, superconduciting qubits have recently enabled the first demonstration of quantum supremacy [2]. In quantum communication and sensing, secure protocols [30] and digital radars [3] have been successfully demonstrated. In addition, the strong light-matter coupling achievable in circuit QED setups (coupling of microwave photons with superconducting qubits) have found new regimes of physics beyond the Jaynes-Cummings model [33], and enabled many hybrid quantum systems [9]. This rising field of circuit QED / microwave quantum optics has its roots inthe more traditional disciplines of condensed matter, atomic and molecular optics (AMO), and nonlinear optics. In this thesis, we find it fruitful to take the more traditional perspectives anduse the familiar languages of nonlinear optics and AMO. We will consistently describe su- perconducting qubits and microwave transmission line waveguides as artificial atoms and metamaterials. We will speak of the nonlinear susceptibility 휒(3), and the “atomic” or “photonic” nature of qubits and waveguides. The analogy between these fields lead to fundamental insights that should hopefully appeal to a broad audience.