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Superconducting Qubits and Circuits: Artificial Atoms Coupled to Microwave Photons Steven M. Girvin1 Yale University Department of Physics New Haven, CT 06520 United States of America ⃝c 2011 1⃝c S.M. Girvin, 2011 Lectures delivered at Ecole d'Et´eLes Houches, July 2011 To be published by Oxford University Press. ii S.M. Girvin These lectures are based on work done in collabora- tion with many students, postdocs and colleagues, including particularly my friends and colleagues Michel H. Devoret and Robert J. Schoelkopf. Contents 0.1 Outline . 1 1 Introduction to Quantum Machines 3 2 Quantum Electrical Circuits 5 2.1 Introduction . 5 2.2 Plasma Oscillations . 6 2.3 Quantum LC Oscillator . 8 2.3.1 Coherent States . 15 2.4 Modes of Transmission Lines Resonators . 18 2.5 `Black Box' Quantization . 23 3 Superconductivity 27 4 Superconducting Qubits 33 4.1 The Cooper Pair Box . 33 4.2 Inductively Shunted Qubits . 43 5 Noise induced Decoherence in Qubit Circuits 49 6 Introduction to Cavity and Circuit QED 59 7 NOBEL SYMPOSIUM TEXT BEGINS HERE 65 7.1 Introduction . 65 7.2 Circuit QED . 66 7.2.1 The Quantronium Qubit . 70 7.2.2 Transmon Qubits . 71 7.2.3 Fluxonium Qubits . 74 7.3 Recent Progress and Future Directions . 74 8 Appendix: Semi-Infinite Transmission Lines, Dissipation and Input/Output Theory 79 8.0.1 Quantum Input-output theory for a driven cavity . 84 Bibliography 93 iii iv S.M. Girvin Circuit QED 1 Key papers students may wish to consult: • Blais et al., `Cavity quantum electrodynamics for superconducting elec- trical circuits: An architecture for quantum computation,' Phys. Rev. A 69, 062320 (2004). • Clerk et al., `Introduction to quantum noise, measurement and am- plification,’ Rev. Mod. Phys (2008) (longer version with pedagogical appendices available at arXiv.org:0810.4729) 0.1 Outline 1. Introduction to Quantum Machines 2. Why are there so few degrees of freedom in SC circuits? (a) Superconducting excitation gap (b) Strong Coulomb force 3. Brief Introduction to the Quantum to Classical Crossover {why is it hard to make large objects that are quantum and not classical, and that have simple spectra like atoms? 4. Elements of a quantum electrical circuit: L,C,JJ 5. Quantization of a lumped-element LC oscillator (a) What is a photon? First quantization and second quantization views. (b) Role of circuit admittance Y (!) (c) Distributed element resonators: Box modes and how to quantize them using admittance 6. Josephson junctions (a) Just what is superconductivity anyway? (b) How does a Josephson junction JJ work? 7. Superconducting Qubits (a) Brief survey of different SC qubit topologies and geometries: charge, phase, flux, fluxonium (b) dephasing and decay mechanisms, T1;T2, sweet spots (c) strengths and weaknesses of different qubit designs (d) The transmon: World's simplest SC qubit: a JJ and an antenna 8. Introduction to Cavity and Circuit QED (a) Jaynes-Cummings model for a two-level atom 2 S.M. Girvin (b) dressed atom/polariton picture (c) vacuum Rabi coupling and degenerate regime (d) dispersive regime (e) strong dispersive regime, number splitting (f) weakly anharmonic transmon in a cavity: perturbation expansion in the anharmonicity; α ∼ g 9. Input-Output Theory (a) qubit driving via driving cavity (b) quantum noise, dissipation, decay and the Purcell effect (c) linear readout, back action, quantum Zeno (d) non-linear readout methods i. Josephson bifurcation amplifier ii. Chirped JBA iii. Using the self-non-linearity induced by qubit 10. Quantum Control and Algorithms (a) single qubit rotations, Rabi, Ramsey (b) two-qubit entangling operations using conditional phase shifts (c) qubit-cavity entangling operations in the strong-dispersive regime (d) QND measurement of photon number (e) Bell/CSHS inequalities (f) Grover Search and Deutsch-Josza algorithms on a two-qubit proces- sor (g) GHZ/Mermin inequalities (h) Towards Quantum Error Correction i. measurement based QEC ii. non-measurement based QEC 11. Towards Scaling up to Many Qubits (a) importance of large on-off coupling ratios (b) Houck double transmon (c) multiple-cavity architectures and communication protocols (d) cavities as quantum memories (e) cavities as qubits (f) qubits as switches (g) Many-body physics with lattice models for strongly interacting po- laritons Chapter 1 Introduction to Quantum Machines A quantum machine is a device whose degrees of freedom are intrinsically quan- tum mechanical. While we do not yet fully understand the properties of such devices, there is great hope (and some mathematical proof) that such machines will have novel capabilities that are impossible to realize on classical hardware. You might think that quantum machines have already been built. For example, the laser and the transistor would seem to rely on quantum physics for their operation. It is clear that the frequency of a laser cannot be computed without the quantum theory that predicts the excitation energies of the atoms in the laser. Similarly the optimal bias voltage of a bipolar transistor depends on the electronic band gap energy of the material from which it is made. Nevertheless, it is only the particular values of the operating parameters of these machines that are determined by quantum physics. Once we know the values of these parameters, we see that these are classical machines because their degrees of freedom are purely classical. Indeed the light output from a laser is special because it is exactly like the RF output of the classical oscillator that powers a radio station's antenna. Similarly, the currents and voltages in an ordinary transistor circuit need not be treated as non-commuting quantum operators in order to understand the operation of the circuit. Its degrees of freedom are, for all intents and purposes, classical. These lectures are devoted to understanding the basic components of quan- tum machines that can be constructed from superconducting electrical circuits. These circuits can be used to create resonators which store individual microwave photons and to create superconducting quantum bits. Both of these circuit ele- ments are intrinsically quantum mechanical. They have quantized energy levels with spacing much greater than temperature (for low enough temperatures!) and they can be placed into quantum superpositions of different energy states. To properly predict their properties, the currents and voltages in such circuits must be represented by non-commuting quantum operators. 3 4 S.M. Girvin The beauty of electrical circuits is that they can be constructed in a modular manner combining together a few different building blocks in a simple manner. The wires connecting these building blocks have to be capable of carrying quan- tum signals [1], but are still relatively simple and the problem of spatial mode matching that occurs in optical circuits is largely eliminated. Furthermore, with modern lithographic techniques, parallel fabrication of complex structures is rel- atively straightforward lending hope that (someday) it will be possible to scale up to processors with large numbers of qubits. We will study different qubit designs and their relative merits. We will also learn how to control and readout the quantum states of qubits and cavities, and how to entangle different their quantum degrees of freedom. In recent decades, we have come to understand that superposition and entanglement are features of quantum physics which can be used as resources to make powerful quantum computers to process information in ways that are impossible classically. What else we can do with quantum machines is not yet fully understood. The people who invented the laser had no idea that it would be used to play music, com- municate over optical fiber, and perform eye surgery. We expect that similar surprises and unexpected applications will be developed once quantum hardware becomes routine enough to play with. Chapter 2 Quantum Electrical Circuits 2.1 Introduction Quantum electrodynamics is the theory of interaction between electrons (and atoms) with electromagnetic fields. These lecture notes discuss the closely re- lated problem of quantization of electrical circuits [1,2]. Experimental progress over the last decade in creating and controlling quantum coherence in supercon- ducting electrical circuits has been truly remarkable. The quantum electrody- namics of superconducting microwave circuits has been dubbed `circuit QED' by analogy to cavity QED in quantum optics. These lecture notes will describe the quantum optics approach to microwave circuits with superconducting qubits playing the role of artificial atoms whose properties can be engineered. Despite being large enough to be visible to the naked eye, these artificial atoms have a very simple discrete set of quantized energy levels which are nearly as well understood as those of the prototypical single-electron atom, hydrogen. Fur- thermore it has proven possible to put these atoms into coherent superpositions of different quantum states so that they can act as quantum bits. Through clever engineering, the coherence times of such superposition states has risen more than four orders of magnitude from nanoseconds for the first supercon- ducting qubit created in 1999 [3] to several tens of microseconds today [4]. This 'Moore's Law' for the exponential growth of coherence time is illustrated in Fig. (??). In addition to being a potentially powerful engineering architecture for build- ing a quantum computer, circuit QED opens up for us a novel new regime to study ultra-strong coupling between `atoms' and individual microwave photons. The concept of the photon is a subtle one, but hopefully these notes will convince the reader that microwaves, despite their name, really are particles. We will ac- cordingly begin our study with a review of the quantization of electromagnetic fields in circuits and cavities. The quantization of electrical circuits has been thoroughly addressed in the Les Houches lecture notes of my colleague, Michel Devoret [2], to which I direct 5 6 S.M.
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