DOCSLIB.ORG

Cavity State Reservoir Engineering in Circuit Quantum Electrodynamics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Cavity State Reservoir Engineering in Circuit Quantum Electrodynamics Cavity State Reservoir Engineering in Circuit Quantum Electrodynamics A Dissertation Presented to the Faculty of the Graduate School of Yale University in Candidacy for the Degree of Doctor of Philosophy by Eric T. Holland Dissertation Director: Robert J. Schoelkopf August 2015 ©2015 – Eric T. Holland all rights reserved. Thesis advisor: Professor Robert J. Schoelkopf Eric T. Holland Cavity State Reservoir Engineering in Circuit Quantum Electrodynamics Abstract Engineered quantum systems are poised to revolutionize information science in the near future. A persistent challenge in applied quantum technology is creat- ing controllable, quantum interactions while preventing information loss to the environment, decoherence. In this thesis, we realize mesoscopic superconducting circuits whose macroscopic collective degrees of freedom, such as voltages and currents, behave quantum mechanically. We couple these mesoscopic devices to microwave cavities forming a cavity quantum electrodynamics (QED) architec- ture comprising entirely of circuit elements. This application of cavity QED is dubbed Circuit QED and is an interdisciplinary field seated at the intersection of electrical engineering, superconductivity, quantum optics, and quantum infor- mation science. Two popular methods for taming active quantum systems in the presence of decoherence are discrete feedback conditioned on an ancillary sys- tem or quantum reservoir engineering. Quantum reservoir engineering maintains a desired quantum state through a combination of drives and designed entropy evacuation. Circuit QED provides a favorable platform for investigating quan- tum reservoir engineering proposals. A major advancement of this thesis is the development of a quantum reservoir engineering protocol which maintains the quantum state of a microwave cavity in the presence of decoherence. This thesis synthesizes of strongly coupled, coherent devices whose solutions to its driven, dissipative Hamiltonian are predicted a priori. This work lays the foundation for future advancements in cavity centered quantum reservoir engineering pro- tocols that have potential to realize hardware efficient protocols to protect more elaborate quantum states. iii Contents List of Figures viii List of Tables x Publication List xi Acknowledgements xii 1 Introduction 1 1.1 Thesis Overview ............................ 5 2 Resonator Theory 7 2.1 LC Circuit ............................... 9 2.2 LCR Circuit .............................. 11 2.3 Quality factors ............................ 12 2.3.1 Quality Factor Parallel LCR Circuit . 13 2.4 Coupling to LC Oscillator ...................... 15 2.5 Participation Ratios ......................... 18 2.5.1 Lossy Capacitor ........................ 18 2.5.2 Partially Filled Lossy Capacitor-Parallel . 20 2.5.3 Partially Filled Lossy Capacitor-Series . 25 2.5.4 Lossy Inductor ........................ 27 2.5.5 Partially Filled Lossy Inductor-Series . 28 2.5.6 Partially Filled Lossy Inductor-Parallel . 30 2.6 Final Thoughts ............................ 33 3 Transmission Line Resonators 34 3.1 Historical Development ........................ 35 3.2 Characteristic Impedance ....................... 36 iv 3.3 Design Consideration ......................... 38 3.4 First Design .............................. 41 3.5 Participation Ratios ......................... 43 3.5.1 Surface Dielectric Loss .................... 44 3.5.2 Bulk Dielectric Loss ..................... 47 3.5.3 Inductive losses and Kinetic Inductance Ratio . 49 3.6 Evanescent Coupling ......................... 52 3.6.1 Asymmetry and Coupling Through TE01 Mode . 57 3.7 Quality Factor Measurements .................... 58 3.8 Kinetic Inductance Ratio Measurements . 61 3.9 Limitations .............................. 62 3.9.1 Coupling ............................ 64 3.9.2 Frequency Stability ...................... 64 3.9.3 Issue Resolution ........................ 65 3.10 Future Outlook ............................ 67 4 Circuit Quantum Electrodynamics 68 4.1 Cavity Quantum Electrodynamics . 71 4.2 Josephson Junction as a Circuit Element . 73 4.3 Circuit Quantization ......................... 76 4.3.1 LC Oscillator ......................... 76 4.3.2 Anharmonic Oscillator: The Transmon . 77 4.4 Transmon Coupled to Harmonic Oscillator . 79 4.4.1 Strong Dispersive Regime . 82 4.5 Black box quantization ........................ 84 5 Experimental Techniques 89 5.1 Fabrication .............................. 90 5.1.1 Substrates ........................... 90 5.1.2 Lithography .......................... 91 5.1.3 Deposition and Dicing .................... 94 5.2 Sample Holders ............................ 95 5.2.1 Striplines ........................... 95 5.2.2 Cavities ............................ 96 5.3 Experimental Setup .......................... 96 5.3.1 Dilution Refrigerator ..................... 96 5.3.2 Vericold Wiring and Filtering . 97 5.3.3 Cavity Filtering ........................ 97 v 5.3.4 Heterodyne Measurement Setup . 101 5.4 Cavity Measurement Techniques . 103 5.4.1 Transmission . 103 5.5 Qubit Measurement Techniques . 105 5.5.1 Dispersive Readout . 107 5.5.2 Jaynes-Cummings Readout . 108 6 Cavity State Reservoir Engineering 111 6.1 Introduction .............................. 111 6.1.1 Measurement with External Feedback . 112 6.1.2 Quantum Reservoir Engineering . 116 6.2 Fock State Stabilization Protocol . 118 6.2.1 Four State Model . 121 6.3 Linblad Master Equation and Simulation . 124 6.4 Photon Number Calibration via the AC stark effect . 125 6.5 Hamiltonian Parameters . 129 6.5.1 Anharmonicities . 131 6.5.2 Cavity-Cavity Cross-Kerr . 133 6.6 Photon Number Selective π Pulse Calibration . 135 6.6.1 π Pulse Selectivity . 137 6.7 Fock State Stabilization Protocol . 137 6.7.1 Comparison DDROP and FSSP . 140 6.7.2 Steady State Wigner Function for Stabilized N=1 Fock State142 6.7.3 Interpreting Results as a Spin System . 144 6.8 Future Improvements . 147 6.8.1 Final Comment . 151 7 Conclusion & Outlook 152 7.1 Conclusion ............................... 152 7.2 Future Experiments . 153 7.2.1 Extension to Fock State Stabilization . 153 7.2.2 N00N State Stabilization . 154 7.3 Future Outlook ............................ 156 Appendix A Recipes 158 A.1 Cavity Photolithography . 158 A.1.1 Resist ............................. 158 A.1.2 Development . 159 A.2 Cavity Etching ............................ 160 vi A.2.1 Tools Required . 160 A.2.2 Setting Up . 161 A.2.3 Acid change at two hour mark . 161 A.2.4 Finish ............................. 162 A.3 Dolan Bridge Josephson Junction Recipe . 163 A.3.1 Wafer Cleaning (15 minutes) . 163 A.3.2 Resist spinning (45 minutes) . 164 A.3.3 Anti-charging Layer (1 hour) . 164 A.3.4 Electron beam writing (2-4 hours depending area to be writ- ten) .............................. 165 A.3.5 Development (15 minutes) . 165 A.3.6 Deposition (4 hours) . 166 A.3.7 Liftoff (2+hours) . 166 A.3.8 Dicing (4 hours) . 167 A.3.9 After dicing (2 hours) . 167 Appendix B QuTiP Code 169 B.1 Fock State Stabilization . 169 B.1.1 Time Dependent Solver . 169 B.1.2 Steady State Solver . 173 Bibliography 189 vii List of Figures 2.1 LC Circuit Diagram ......................... 9 2.2 LCR Circuit Diagram ......................... 11 2.3 Coupling to LC Oscillator ...................... 15 2.4 Lossy Capacitor ............................ 18 2.5 Lossy Capacitor-Partially Filled Parallel . 21 2.6 Lossy Capacitor-Partially Filled Series . 24 2.7 Lossy Inductor ............................ 27 2.8 Lossy Inductor-Partially Filled Series . 29 2.9 Lossy Inductor-Partially Filled Parallel . 31 3.1 Microstrip Tri-Plate and Stripline . 37 3.2 Prototypical Stripline ......................... 38 3.3 Rectangular Waveguide Cavity .................... 40 3.4 Stripline First Design ......................... 42 3.5 Center Conductor Surface Participation Ratio . 48 3.6 Kinetic Inductance Calculation ................... 51 3.7 Evanescent Coupling to 3D Cavity and Stripline . 56 3.8 Total Quality Factor versus Temperature . 59 3.9 Measured Kinetic Inductance of Stripline Resonator and Holder . 62 3.10 Stripline Second Design ........................ 63 3.11 Coaxline ................................ 66 4.1 Jaynes-Cummings Model ....................... 70 4.2 CQED Realization .......................... 72 4.3 Josephson Junction Schematic and Circuit Equivalent . 73 4.4 Circuit Representation of Transmon . 78 4.5 Circuit Representation of Transmon Coupled to LC Oscillator . 81 4.6 Charge Based Superconducting Qubits . 83 4.7 Black Box Quantization HFSS Simulation . 86 viii 5.1 Double Angle Dolan Bridge Deposition . 93 5.2 Vericold Dilution Refrigerator Wiring . 98 5.3 Room Temperature Microwave Control and Data Acquisition . 102 5.4 Dispersive Readout . 106 5.5 Jaynes-Cummings Readout . 109 6.1 Conceptual schematic of reservoir engineering . 116 6.2 Fock state stabilization protocol . 120 6.3 Four state model . 123 6.4 AC Stark Photon Number Calibration . 126 6.5 AC Stark Amplitude . 129 6.6 AC Stark Calibration . 130 6.7 Storage Cavity Anharmonicity . 132 6.8 Single Photon Cross-Kerr . 133 6.9 Photon Number Poisson Distribution . 138 6.10 P(0) and P(1) ............................. 141 6.11 Steady State Wigner function . 145 6.12 Steady State Cavity Polarization . 148 6.13 Expected FSS ............................. 150 7.1 Historical Progress of Superconducting Circuits . 156 ix List of Tables 6.1 Fock State Stabilization Parameters . 135 6.2 Comparison FSSP and DDROP . 142 x Publication List This thesis is based on the following
Load more

Recommended publications
  • The Quantum Toolbox in Python Release 2.2.0
    QuTiP: The Quantum Toolbox in Python Release 2.2.0 P.D. Nation and J.R. Johansson March 01, 2013 CONTENTS 1 Frontmatter 1 1.1 About This Documentation.....................................1 1.2 Citing This Project..........................................1 1.3 Funding................................................1 1.4 Contributing to QuTiP........................................2 2 About QuTiP 3 2.1 Brief Description...........................................3 2.2 Whats New in QuTiP Version 2.2..................................4 2.3 Whats New in QuTiP Version 2.1..................................4 2.4 Whats New in QuTiP Version 2.0..................................4 3 Installation 7 3.1 General Requirements........................................7 3.2 Get the software...........................................7 3.3 Installation on Ubuntu Linux.....................................8 3.4 Installation on Mac OS X (10.6+)..................................9 3.5 Installation on Windows....................................... 10 3.6 Verifying the Installation....................................... 11 3.7 Checking Version Information via the About Box.......................... 11 4 QuTiP Users Guide 13 4.1 Guide Overview........................................... 13 4.2 Basic Operations on Quantum Objects................................ 13 4.3 Manipulating States and Operators................................. 20 4.4 Using Tensor Products and Partial Traces.............................. 29 4.5 Time Evolution and Quantum System Dynamics.........................
    [Show full text]
  • Jupyter Notebooks—A Publishing Format for Reproducible Computational Workflows
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elpub digital library Jupyter Notebooks—a publishing format for reproducible computational workflows Thomas KLUYVERa,1, Benjamin RAGAN-KELLEYb,1, Fernando PÉREZc, Brian GRANGERd, Matthias BUSSONNIERc, Jonathan FREDERICd, Kyle KELLEYe, Jessica HAMRICKc, Jason GROUTf, Sylvain CORLAYf, Paul IVANOVg, Damián h i d j AVILA , Safia ABDALLA , Carol WILLING and Jupyter Development Team a University of Southampton, UK b Simula Research Lab, Norway c University of California, Berkeley, USA d California Polytechnic State University, San Luis Obispo, USA e Rackspace f Bloomberg LP g Disqus h Continuum Analytics i Project Jupyter j Worldwide Abstract. It is increasingly necessary for researchers in all fields to write computer code, and in order to reproduce research results, it is important that this code is published. We present Jupyter notebooks, a document format for publishing code, results and explanations in a form that is both readable and executable. We discuss various tools and use cases for notebook documents. Keywords. Notebook, reproducibility, research code 1. Introduction Researchers today across all academic disciplines often need to write computer code in order to collect and process data, carry out statistical tests, run simulations or draw figures. The widely applicable libraries and tools for this are often developed as open source projects (such as NumPy, Julia, or FEniCS), but the specific code researchers write for a particular piece of work is often left unpublished, hindering reproducibility. Some authors may describe computational methods in prose, as part of a general description of research methods.
    [Show full text]
  • Arxiv:1110.0573V2 [Quant-Ph] 22 Nov 2011 Equation Or Monte-Carlo Wave Function Method
    QuTiP: An open-source Python framework for the dynamics of open quantum systems J. R. Johanssona,1,∗, P. D. Nationa,b,1,∗, Franco Noria,b aAdvanced Science Institute, RIKEN, Wako-shi, Saitama 351-0198, Japan bDepartment of Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA Abstract We present an object-oriented open-source framework for solving the dynamics of open quantum systems written in Python. Arbitrary Hamiltonians, including time-dependent systems, may be built up from operators and states defined by a quantum object class, and then passed on to a choice of master equation or Monte-Carlo solvers. We give an overview of the basic structure for the framework before detailing the numerical simulation of open system dynamics. Several examples are given to illustrate the build up to a complete calculation. Finally, we measure the performance of our library against that of current implementations. The framework described here is particularly well-suited to the fields of quantum optics, superconducting circuit devices, nanomechanics, and trapped ions, while also being ideal for use in classroom instruction. Keywords: Open quantum systems, Lindblad master equation, Quantum Monte-Carlo, Python PACS: 03.65.Yz, 07.05.Tp, 01.50.hv PROGRAM SUMMARY 1. Introduction Manuscript Title: QuTiP: An open-source Python framework for the dynamics of open quantum systems Every quantum system encountered in the real world is Authors: J. R. Johansson, P. D. Nation an open quantum system [1]. For although much care is Program Title: QuTiP: The Quantum Toolbox in Python taken experimentally to eliminate the unwanted influence Journal Reference: of external interactions, there remains, if ever so slight, Catalogue identifier: a coupling between the system of interest and the exter- Licensing provisions: GPLv3 nal world.
    [Show full text]
  • Qutip: Quantum Toolbox in Python
    Lecture on QuTiP: Quantum Toolbox in Python with case studies in Circuit-QED Robert Johansson RIKEN In collaboration with Paul Nation Korea University Pohang 2012 [email protected] 1 Content ● Introduction to QuTiP ● Case studies in circuit-QED – Jaynes-Cumming-like models ● Vacuum Rabi oscillations ● Qubit-gates using a resonators as a bus ● Single-atom laser – Dicke model / Ultrastrong coupling – Correlation functions and nonclassicality tests – Parametric amplifier Pohang 2012 [email protected] 2 resonator as coupling bus qubits qubit-qubit NIST 2007 UCSB 2012 UCSB 2009 NIST 2002 high level of control of resonators UCSB 2009 Yale 2011 Yale 2004 Delft 2003 qubit-resonator Saclay 1998 Yale 2008 ETH 2010 UCSB 2006 NEC 1999 Saclay 2002 Chalmers 2008 NEC 2003 NEC 2007 ETH 2008 2000 2005 2010 Pohang 2012 [email protected] 3 What is QuTiP? ● Framework for computational quantum dynamics – Efficient and easy to use for quantum physicists – Thoroughly tested (100+ unit tests) – Well documented (200+ pages, 50+ examples) – Quite large number of users (>1000 downloads) ● Suitable for – theoretical modeling and simulations – modeling experiments ● 100% open source ● Implemented in Python/Cython using SciPy, Numpy, and matplotlib Pohang 2012 [email protected] 4 Project information Authors: Paul Nation and Robert Johansson Web site: http://qutip.googlecode.com Discussion: Google group “qutip” Blog: http://qutip.blogspot.com Platforms: Linux and Mac License: GPLv3 Download: http://code.google.com/p/qutip/downloads Repository: http://github.com/qutip Publication: Comp. Phys. Comm. 183, 1760 (2012) Pohang 2012 [email protected] 5 What is Python? Python is a modern, general-purpose, interpreted programming language Modern Good support for object-oriented and modular programming, packaging and reuse of code, and other good programming practices.
    [Show full text]
  • Arxiv:1807.11731V2
    QEngine: An open-source C++ Library for Quantum Optimal Control of Ultracold Atoms J. J. W. H. Sørensena, J. H. M. Jensena, T. Heinzela, J. F. Shersona aDepartment of Physics and Astronomy, Aarhus University, Ny Munkegade 120, 8000 Aarhus C Denmark Abstract We present the first version of the QEngine, an open-source C++ library for simulating and controlling ultracold quantum systems using optimal control theory (OCT). The most notable systems presented here are Bose-Einstein condensates, many-body systems described by Bose-Hubbard type models, and two inter- acting particles. These systems can all be realized experimentally using ultracold atoms in various trapping geometries including optical lattices. In addition we provide a number of optimal control algorithms includ- ing the recently introduced group method. The QEngine library has a strong focus on accessibility and performance. We provide several examples of how to prepare simulations of the physical systems and apply optimal control. Keywords: Quantum Optimal Control Theory; Bose-Einstein Condensate; Gross-Pitaevskii equation; group; Bose-Hubbard; C++. PROGRAM SUMMARY Program Title: QEngine Program Summary: quatomic.com Download : gitlab.com/quatomic/qengine License: MPL-2.0 Programming Language: C++14 Computer: Any system with a C++14 compliant compiler. Operating system: Linux, Mac OSX, Windows RAM : 2+ Gigabytes External routines: Armadillo, LAPACK and BLAS or Intel Math Kernel Library Nature of Problem: Quantum optimal control of ultracold systems. Solution method: Numerical simulation of the equation of motion and gradient based quantum control. Running time: Few seconds up to several hours depending on the size of the underlying Hilbert space. 1.
    [Show full text]
  • PDF Documentation
    QuTiP: Quantum Toolbox in Python Release 4.3.0 P.D. Nation, J.R. Johansson, A.J.G. Pitchford, C. Granade, and A.L. Grimsmo Feb 26, 2019 Contents 1 Frontmatter 3 1.1 About This Documentation.....................................3 1.2 Citing This Project..........................................3 1.3 Funding................................................3 1.4 About QuTiP.............................................4 1.5 Contributing to QuTiP........................................5 2 Installation 7 2.1 General Requirements........................................7 2.2 Platform-independent Installation..................................7 2.2.1 Building your Conda environment.............................8 2.2.2 Adding the conda-forge channel..............................8 2.3 Installing via pip...........................................8 2.4 Installing from Source........................................9 2.5 Installation on MS Windows.....................................9 2.5.1 Windows and Python 2.7.................................. 10 2.6 Verifying the Installation....................................... 10 2.7 Checking Version Information using the About Function...................... 10 3 Users Guide 11 3.1 Guide Overview........................................... 11 3.1.1 Organization........................................ 11 3.2 Basic Operations on Quantum Objects................................ 12 3.2.1 First things first....................................... 12 3.2.2 The quantum object class.................................. 13 3.2.3
    [Show full text]
  • Teaching Quantum Mechanics with Python
    22 for submission to the Board of Trustees for action. The Program Director may All letters of inquiry and completed formal applications should be mailed in request additional information, an interview with the applicant,TEACHING or a visit to hard copy to: QUANTUM the applicant’s organization. The full proposal, including staff summary and John Van Zytveld, Ph.D. analysis, is made available to the Trustees for their consideration and decision. Senior Program Director The applicant is notified promptly when a decision has been reached. While M. J. Murdock Charitable Trust some level of merit is evident in nearly everyMECHANICS proposal received by the Trust, only P. O. Box 1618 WITH PYTHON a fraction of the requests reviewed can result in awards. When an application has Vancouver, WA 98668 been declined, it will not be carried over for future consideration. Under normal AndrewFor More Help M.C. Dawes circumstances, re-submission of a proposal that was declined is not encouraged. If your questions have not been answered by this document or you need some Each proposal becomes the property of the Trust and will not be returned. It will additional information, please call us at 360.694.8415. be treated as a privileged communication with the understanding, however,@drdawes that amcdawes.com it may be peer reviewed. Mailing Address: https://github.com/amcdawes/QMlabsM. J. Murdock Charitable Trust PO Box 1618 Vancouver, Washington 98668 Office Location: M. J. Murdock Executive Plaza 703 Broadway, Suite 710 Vancouver, Washington 98660 Contact:
    [Show full text]
  • Robust Ion Trap Quantum Computation Enabled by Quantum Control
    Robust Ion Trap Quantum Computation Enabled by Quantum Control by Pak Hong (James) Leung Department of Physics Duke University Date: Approved: Kenneth Brown, advisor Stephen Teitsworth Harold Baranger Thomas Barthel Jungsang Kim Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Physics in the Graduate School of Duke University 2020 ABSTRACT Robust Ion Trap Quantum Computation Enabled by Quantum Control by Pak Hong (James) Leung Department of Physics Duke University Date: Approved: Kenneth Brown, advisor Stephen Teitsworth Harold Baranger Thomas Barthel Jungsang Kim Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Physics in the Graduate School of Duke University 2020 Copyright by Pak Hong (James) Leung 2020 Abstract The advent of quantum computation foretells a new era in science and technology, but the fragility of quantum bits (qubits) and the unreliability of gates hinder the realization of functioning quantum computers. For ion trap quantum computers in particular, 2-qubit operations relying on the Mølmer-Sørensen interaction have the greatest error rates. This dissertation introduces frequency-modulated (FM) pulses as a measure to maximize 2-qubit gate fidelity and a way to calibrate gate errors through the measurement of circuit performance. A key challenge of two-qubit gates in ion chains is unwanted residual entanglement between the ion spin and its motion. Frequency-modulated pulses are developed to achieve such goal. This theoretical advance has led to high-fidelity 2-qubit gates that are robust against small frequency drifts in a 5-ion experiment.
    [Show full text]
  • Qutip: a Framework for the Dynamics of Open Quantum Systems Using Scipy and Cython
    56 PROC. OF THE 11th PYTHON IN SCIENCE CONF. (SCIPY 2012) QuTiP: A framework for the dynamics of open quantum systems using SciPy and Cython ‡ ‡ Robert J. Johansson ∗, Paul D. Nation QuTiP Organization 6/14/12 3:35 PM F Abstract—We present QuTiP (http://www.qutip.org), an object-oriented, open- three_level_basis three_level_ops source framework for solving the dynamics of open quantum systems. Written wigner in Python, and using a combination of Cython, NumPy, SciPy, and matplotlib, qfunc tensor liouvillian steadystate QuTiP provides an environment for computational quantum mechanics that about spost spre Bloch brmesolve bloch_redfield_tensor is both easy and efficient to use. Arbitrary quantum systems, including time- thermal_dmsteady clebsch qutrit_basis three_level_atom correlation spectrum_ss dependent systems, may be built up from operators and states defined by a projection demos ket2dm superoperator concurrence quantum object class, and then passed on to a choice of unitary or dissipative fock_dm entropy_conditional coherent_dm wigner entropy_linear tensor entropy_mutual evolution solvers. Here we give an overview of the basic structure for the fock entropy_vn coherent concurrence steady about Bloch bloch-redfield framework, and the techniques used in its implementation. We also present basis clebsch eseries correlation sphereplot demos a few selected examples from current research on quantum mechanics that states essolve entropy ode2es sp_eigs illustrate the strengths of the framework, as well as the types of calculations sphereplot eseries expect simdiag essolve that can be performed. This framework is particularly well suited to the fields of sparse file_data_read expect file_data_store rhs_generate simdiag rhs_generate fileio qload quantum optics, superconducting circuit devices, nanomechanics, and trapped qsave rand_unitary rand floquet ions, while also being ideal as an educational tool.
    [Show full text]
  • Lecture 7: Quantum Master Equations
    Lecture 7: Quantum master equations Hendrik Weimer Institute for Theoretical Physics, Leibniz University Hannover Quantum Physics with Python, 13 June 2016 Outline of the course 1. Introduction to Python 2. SciPy/NumPy packages 3. Plotting and fitting 4. QuTiP: states and operators 5. Ground state problems 6. Time evolution and quantum quenches 7. Quantum master equations 8. Generation of squeezed states 9. Quantum computing 10. Grover's algorithm and quantum machine learning 11. Student presentations Hendrik Weimer (Leibniz University Hannover) Lecture 7: Quantum master equations Open quantum systems I Quantum systems are never perfectly isolated from the outside world I Description in terms of pure state vectors no longer holds I Dynamics no longer described by the unitary Schr¨odingerequation Hendrik Weimer (Leibniz University Hannover) Lecture 7: Quantum master equations The density operator I Statistical mixture of pure quantum states X ρ = pij iih ij i I Proability pi to find the system in the pure state j ii I Conservation of probability: Trfρg = 1, positivity: ρ ≥ 0. I Unitary part of the Dynamics: Liouville-von Neumann equation: d d X X d d ρ = p j ih j = p j i h j + j i h j dt dt i i i i dt i i i dt i i i X i i = − pi (Hj iih ij − j iih ijH) = − [H; ρ] i ~ ~ Hendrik Weimer (Leibniz University Hannover) Lecture 7: Quantum master equations Reduced density matrices I Biparite system (parts A and B) X X j i = cijjiiAjjiB ≡ cijjiji ij ij I Reduced density matrices: partial trace X X 0 0 ρA = TrBfρg = hijjρji jijiihi j i;i0 j I Pure states: X X ∗ 0 ρA = cijci0jjiihi j i;i0 j Hendrik Weimer (Leibniz University Hannover) Lecture 7: Quantum master equations Example: Bell states Qobj.ptrace(list) list contains the parts that should be kept import numpy as np from qutip import basis, tensor psi = (tensor(basis(2, 0), basis(2, 0)) +\ tensor(basis(2, 1), basis(2, 1)))/np.sqrt(2) print(psi.ptrace([0])) Output: Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isherm = True Qobj data = [[ 0.5 0.
    [Show full text]
  • Qutip: Quantum Toolbox in Python Release 4.0.2
    QuTiP: Quantum Toolbox in Python Release 4.0.2 P.D. Nation, J.R. Johansson, A.J.G. Pitchford, C. Granade, and A.L. Grimsmo Jan 11, 2017 Contents 1 Frontmatter 3 1.1 About This Documentation.....................................3 1.2 Citing This Project..........................................3 1.3 Funding................................................3 1.4 About QuTiP.............................................4 1.5 Contributing to QuTiP........................................5 2 Installation 7 2.1 General Requirements........................................7 2.2 Platform-independent Installation..................................7 2.2.1 Building your Conda environment.............................8 2.2.2 Adding the conda-forge channel..............................8 2.3 Installing via pip...........................................9 2.4 Installing from Source........................................9 2.5 Verifying the Installation.......................................9 2.6 Checking Version Information using the About Function......................9 2.7 Installation on MS Windows..................................... 10 3 Users Guide 11 3.1 Guide Overview........................................... 11 3.1.1 Organization........................................ 11 3.2 Basic Operations on Quantum Objects................................ 11 3.2.1 First things first....................................... 11 3.2.2 The quantum object class.................................. 13 3.2.3 Functions operating on Qobj class............................. 17
    [Show full text]
  • Experiments on Superconducting Qubits Coupled to Resonators
    Epeiets o Supeoduig Quits Coupled to Resoatos Utesuhug a Resoatoe gekoppelte supaleitede Quits Zur Erlangung des akademischen Grades eines DOKTORS DER NATURWISSENSCHAFTEN von der Fakultät für Physik des Karlsruher Institut für Technologie (KIT) genehmigte DISSERTATION von Dipl.-Phys. Markus Jerger aus Bühl Tag der mündlichen Prüfung: 1. Februar 2013 Referent: Prof. Dr. Alexey V. Ustinov Korreferent: Prof. Dr. ir. J. E. Mooij Cotets Itoduio Buildig Bloks 2.1 Superconductivity and the Josephson Effect . 7 2.2 The Flux Qubit . 9 2.2.1 Basic Types of Superconducting Qubits . 9 2.2.2 Classical Description of the Flux Qubit . 11 2.2.3 Quantum-Mechanical Description . 14 2.2.4 Truncation to two Levels . 16 2.2.5 Driven Qubit . 18 2.2.6 Bloch Sphere . 20 2.2.7 Qubit in a Dissipative Environment . 21 2.3 Microwave Resonators . 24 2.3.1 Series and Parallel Resonant Circuits . 24 2.3.2 Transmission Lines . 26 2.3.3 Transmission Line Resonators . 28 2.3.4 Circuit Characterization using S Parameters . 31 2.3.5 Resonator Characterization . 33 2.3.6 Parameters of Superconducting Transmission Lines 35 2.4 Circuit Quantum Electrodynamics . 38 2.4.1 Interaction of Light and Matter . 38 2.4.2 Coupling of Resonators to Flux Qubits . 39 2.4.3 The Jaynes-Cummings Hamiltonian . 41 2.4.4 Dispersive Readout . 43 2.5 Frequency-Division Multiplexing . 47 2.5.1 Comparison of Multiplexing Techniques . 47 2.5.2 Multiplexed Dispersive Readout System . 50 2.5.3 Crosstalk and Minimum Channel Bandwidth . 50 i Contents Epeietal Setup ad Tehiue 3.1 Sample Design and Fabrication .
    [Show full text]