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Quantum Feedback for Measurement and Control Quantum feedback for measurement and control by Leigh Martin A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate Division of the University of California, Berkeley Committee in charge: arXiv:2004.09766v1 [quant-ph] 21 Apr 2020 Professor Irfan Siddiqi, Co-chair Professor Birgitta Whaley, Co-chair Professor Hartmut H¨affner Professor Eric Neuscamann Summer 2019 Quantum feedback for measurement and control Copyright 2019 by Leigh Martin 1 Abstract Quantum feedback for measurement and control by Leigh Martin Doctor of Philosophy in Physics University of California, Berkeley Professor Irfan Siddiqi, Co-chair Professor Birgitta Whaley, Co-chair The standard quantum formalism introduced at the undergraduate level treats measurement as an instantaneous collapse. In reality however, no physical process can occur over a truly infinitesimal time interval. A more subtle investigation of open quantum systems lead to the theory of continuous measurement and quantum trajectories, in which wave function col- lapse occurs over a finite time scale associated with an interaction. Within this formalism, it becomes possible to ask many new questions that would be trivial or even ill-defined in the context of the more basic measurement model. In this thesis, we investigate both theoret- ically and experimentally what fundamentally new capabilities arise when an experimental apparatus can resolve the continuous dynamics of a measurement. Theoretically, we show that when one can perform feedback operations on the timescale of the measurement process, the resulting tools provide significantly more control over entanglement generation, and in some settings can generate it optimally. We derive these results using a novel formalism which encompasses most known quantum feedback protocols. Experimentally, we show that continuous measurement allows one to observe the dynamics of a system undergoing simul- taneous non-commuting measurements, which provides a reinterpretation of the Heisenberg uncertainty principle. Finally, we combine the theoretical focus on quantum feedback with the experimental capabilities of superconducting circuits to implement a feedback controlled quantum amplifier. The resulting system is capable of adaptive measurement, which we use to perform the first canonical phase measurement. i To my parents, my step parents, and my amazing sister Willow. ii Contents Contents ii List of Figures iv List of Tables xiii 1 The Building Blocks of Quantum Devices: Qubits, Harmonic Oscillators and Quantum Optics 1 1.1 Introduction . 1 1.2 Qubits and Harmonic Oscillators . 4 1.3 Quantum Optics: The Jaynes-Cummings Hamiltonian . 18 1.4 Applications of Quantum Systems . 23 1.5 Recommended Reading . 27 2 Introduction to circuit QED 30 2.1 Introduction . 30 2.2 Superconductivity: Broad Brush Strokes . 31 2.3 Quantization of a circuit . 39 2.4 Artificial Atoms . 44 2.5 Microwave amplification and detection . 54 3 Open Quantum Systems and Measurement Theory 69 3.1 The POVM Formalism and Weak Measurement . 71 3.2 A Toy Model of Weak Measurement . 74 3.3 Markovian Baths, Continuous Measurement and the Stochastic Master Equa- tions . 86 3.4 Quantum Non-Demolition Measurement and Circuit QED Readout . 91 4 Measurement as a Control Resource 97 4.1 Control With Measurement Alone: Multi-Qubit Gates Using the Zeno Effect 97 4.2 General Quantum Feedback Protocols . 106 4.3 Bi- and Multipartite Entanglement Generation . 110 iii 5 Simultaneous Measurement of Non-Commuting Observables 120 5.1 Achieving high quantum efficiency measurements . 121 5.2 Single and multi-quadrature measurements . 127 5.3 Dynamics under incompatible measurements and a connection to the uncer- tainty principle . 135 5.4 The Polaron Transformation . 139 6 Adaptive Measurements and the Canonical Phase Measurement 142 6.1 Implementation and Verification of a Canonical Phase Measurement . 145 6.2 Adaptive Measurements and Quantum Feedback . 151 7 Optimal measurement-based control 155 7.1 Optimal Entanglement Generation . 156 7.2 Optimal Hamiltonian-Free Control . 168 8 Outlook 172 Bibliography 176 iv List of Figures 1.1 (a) An imperfectly isolated quantum system quickly shares information with its environment. An external detector typically acquires only a small fraction of the total available information, and hence sees no residual coherence during the measurement process. (b) In highly controlled systems such as superconducting circuits, one can arrange that the system interacts most strongly with our detec- tor, allowing one to resolve wave function collapse as a continuous-time process. 2 1.2 (a). Precession of a qubit in the lab frame. The direction of rotation may be obtained by the right-hand rule, and the angular frequency is !q. (b). Rabi oscillations about the σx axis (δ = 0) in the rotating frame. The angular frequency is ΩR = gα0=2. (c). Rabi oscillations for a detuned drive (∆ > ΩR), shown in the frame of the frame of the classical drive field !d................. 8 ~ ~ 1.3 A plot of the z component of the Bloch vector vs. time with H and HRWA. The exact solution oscillates rapidly around the approximate solution but never deviates far from it. Parameters for the graphic are ΩR=2π = 0:5 MHz and !q=2π = 20 MHz. In most realistic systems, the separation in scales between ΩR and !q is even greater. 9 1.4 (a) Rabi oscillations. A best-fit line and the corresponding experimental pulse sequence implemented to measure them. To measure a single point on this graph, one drives the qubit for a duration τ and then measures in the σz basis. Each point is an average hσzi over many runs. Measurement occurs at the `X'. (b) The electric field profile and envelope of a square and a Gaussian pulse. (c) Similar to (a) but a T1 measurement. (d) Ramsey oscillation and the associated pulse sequence. Inset Bloch spheres visualize the qubit dynamics at various points on the trace. The oscillation frequency is set by the qubit-pulse detuning, and the ∗ exponential decay is the coherence time T2 . Best fit indicates an exponential decay time of 7:3 µs.................................. 12 1.5 Q functions for (a) the ground/vacuum state (b) a coherent state (c) the n = 1 Fock state (d) the n = 2 Fock state. 17 v 1.6 The level structure for the Jaynes Cummings Hamiltonian. In the rotating wave approximation, only pairs of levels with the same total excitation number interact, as illustrated with the blue ellipses. Due to the increasing second moment of states as we move up the harmonic oscillator ladder, the coupling strength increases with n. ........................................... 20 1.7 (a) Graphical illustration of two logical qubits each encoded into 3 physical qubits (black boxes). A transversal gate operates as shown. (b) A non-transversal gate. (c) Another non-transversal gate. 26 2.1 (a) A Josephson junction (typically denoted as an X on a circuit diagram) in parallel with a capacitor and a resistor. (b) A mass coupled to a pendulum, which has the same Hamiltonian as (a). A SQUID threaded with an external magnetic flux φext. (d) A physical analog of the SQUID for φext = 0 (top) and φext = φ0=2 (bottom). In the latter case, the gravitational potential of the two masses cancel, effectively removing the junction from the circuit. 37 2.2 (a) A parabolic potential with equally spaced transitions (solid lines) and a quadratic-plus-quartic potential with unequal spacing. Transitions may be ad- dressed individually in the anharmonic case, as they now differ in frequency. (b) Illustration of the integrals used to define branch flux (vertical curved path) and branch charge (loop) of a general circuit element. (c) progressively shrinking an LC circuit until it becomes a microwave cavity. 39 2.3 (a) An opened dilution refrigerator. Stages are cooled by 3 different mechanism. The 4 K plate is cooled by expansion cooling of Helium via a pulse tube cooler. A circuit of a He3/He4 mixture is pumped into the dilution unit and then pumped out of the still with a turbo pump (not pictured). The turbo pump cools the still via evaporative cooling of liquid He3, which has a higher vapour pressure than He4. (b) The principle of dilution refrigeration, which is analogous to evaporative cooling. He4 is bosonic, and has a smaller de Broglie wavelength than He3 due to its higher mass. Both of these properties allow a smaller interatomic spacing between He4 atoms, which in turn leads to a lower energy via the Van der Waals force. Therefore at low temperatures, He3 and He4 phase-separate like oil and water, as depicted. However a He3 atom can lower its energy by entering the He4 phase. Due to the Pauli exclusion principle, He3 atoms entering the He4 phase must occupy increasingly higher energy states. Eventually the Fermi energy exceeds the energy gained by entering the He4 phase. This balance occurs when the He4 phase contains 6.4% He3, even at zero temperature! This allows finite cooling power all the way down to T = 0 (unlike evaporative cooling, where the vapor pressure goes to zero below the boiling point). (c) Circulation of helium during the initial precooling of the fridge (red), used to go from room temperature to about 10 K using the pulse tube, and dilution cooling (blue). 46 vi 2.4 (a) A light-tight can for housing superconducting circuits. DC and microwave control lines enter through tightly sealed bulkheads. A venting hole is necessary to allow evacuation before cooling to mK temperatures. (b) A classic example of a ground loop. Ground loops can be shrunk by running all equipment from the same breaker and ensuring electrical isolation. 47 2.5 (a) The lumped-element circuit for a Cooper pair box (b) The energy levels of a Cooper pair box in the 2-level approximation. The sweet spot occurs at ng = 1=2.
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