Introduction to Quantum Technologies

Preface
Contents
Part IQuantum States and Quantum Mechanics
1 Introduction
	1.1 Common Language in Quantum Information
	1.2 Various Quantum Systems
	1.3 Electromagnetic Waves for Quantum Operations
		1.3.1 Development of Electromagnetic Wave Source
	1.4 Concept of Temperature
2 Linear Algebra
	2.1 Vector Space
		2.1.1 Vectors in Three-Dimensional Vector Space
		2.1.2 Inner Product
		2.1.3 Orthonormal Basis
		2.1.4 Vector Components
		2.1.5 Norm of a Vector
		2.1.6 Outer Product
		2.1.7 Expansion for Multidimensional System
	2.2 Matrix and Operator
		2.2.1 Matrix Element
		2.2.2 Transpose Matrix
		2.2.3 Matrix Multiplication
		2.2.4 Square Matrix
	2.3 Eigenvectors and Eigenvalues
	2.4 Summary of Vector Characteristics in Index Notation
3 Wavefunction and Notations in Quantum Mechanics
	3.1 Equation of Motion in Classical and Quantum Mechanics
	3.2 Wavefunction
		3.2.1 Inner Product of Wavefunction
		3.2.2 Continuous and Discretized Wavefunction
	3.3 Operator
	3.4 Dirac Notation
	3.5 Matrix Representation
	3.6 Properties of Wavefunction
	3.7 Composite System
		3.7.1 Operator in Composite System
	3.8 Examples of Notation for Frequently Used Quantum State
	3.9 Quantum State Representation: Qubit
		3.9.1 One-Qubit state
		3.9.2 Bloch Sphere
		3.9.3 Two-Qubit State
	3.10 Eigenvalue Equation in Quantum Mechanics
		3.10.1 Example: Atom
	3.11 Operator Classification
		3.11.1 Operator Functions
		3.11.2 Hermitian Operator
		3.11.3 Projection Operator
		3.11.4 Unitary Operator
		3.11.5 Pauli Operators
	3.12 Density Operator
		3.12.1 Example of a Mixed State
		3.12.2 General Properties of Density Operator
		3.12.3 Density Operator of Composite System and its Reduction
	3.13 Commutator and Anti-Commutator
		3.13.1 Heisenberg\'s Uncertainty Principle
4 Time Evolution in Quantum System
	4.1 Time-Independent Schrödinger Equation
	4.2 Time Evolution in Terms of Unitary Operator
	4.3 Three Pictures in Quantum Mechanics
		4.3.1 General Overview
		4.3.2 Schrödinger Picture
		4.3.3 Heisenberg Picture
		4.3.4 Example: Harmonic Oscillator in Schrödinger and Heisenberg Picture
		4.3.5 Dirac Picture
	4.4 Heisenberg Equation of Motion
	4.5 von Neumann Equation
	4.6 Unitary Transformation to a Rotating Frame
	4.7 Driven Two-Level System
5 Perturbation Theory
	5.1 Time Independent Perturbation Theory
		5.1.1 Zeeman Effect
	5.2 Treatment of Time-Dependent Perturbation
		5.2.1 Time-Dependent Perturbation Expansion
Part IIHarmonic Oscillator, Qubit and Coupled Quantum Systems
6 Harmonic Oscillator
	6.1 Harmonic Oscillator and Its Hamiltonian
	6.2 Electromagnetic Waves and Ladder Operators
		6.2.1 Quantization of Electromagnetic Waves
		6.2.2 Ladder Operators of a Harmonic Oscillator
	6.3 Quantum States of a Harmonic Oscillator
		6.3.1 Coherent State
		6.3.2 Schrödinger\'s Cat State
		6.3.3 Squeezed State
		6.3.4 Thermal State
	6.4 Photon Correlations
		6.4.1 Amplitude Correlation/First-Order Coherence
		6.4.2 Intensity Correlation/Second-Order Coherence
	6.5 Wigner Function
		6.5.1 General Remarks
		6.5.2 Examples
7 Two-level System and Interaction with Electromagnetic Waves
	7.1 Two-level System, Spin, and Bloch Sphere
	7.2 Interaction Between Two-level System and Electromagnetic Field
		7.2.1 Spontaneous and Stimulated Emission
	7.3 Two-level Systems and Electromagnetic Waves in Practice
		7.3.1 Two-level Systems in Nuclear Magnetic Resonance
		7.3.2 Two-level Systems in Atomic Gases
		7.3.3 Two-level Systems in Solid-state Quantum Defects
		7.3.4 Two-level System in an Optically Controlled Semiconductor Quantum Dot
		7.3.5 Two-level System in Superconducting Circuit
	7.4 Dynamics and Relaxations of Two-level Systems
		7.4.1 Bloch Equations and Relaxations
		7.4.2 Rabi Oscillation
		7.4.3 Ramsey Interference
		7.4.4 Spin Echo
8 Electromagnetic Cavities and Cavity Quantum Electrodynamics
	8.1 Properties of Cavities
		8.1.1 Q Factor
		8.1.2 Finesse
	8.2 Measurement of a Cavity
		8.2.1 Reflection and Transmission Measurements
		8.2.2 Actual Measurement Systems
	8.3 Input-Output Theory
		8.3.1 Propagating Mode and Input-Output Theory
		8.3.2 Single-Port Measurement
		8.3.3 Dual-Port Measurement
	8.4 Cavity Quantum Electrodynamics
		8.4.1 Jaynes–Cummings Model
		8.4.2 Tavis–Cummings Model
		8.4.3 Weak and Strong Coupling Regimes
		8.4.4 Dispersive Regime
		8.4.5 Waveguide-Coupled Cavity QED System
9 Various Couplings in Quantum Systems
	9.1 Interaction Hamiltonians
		9.1.1 Jaynes–Cummings and Anti-Jaynes–Cummings Interactions
		9.1.2 Beam-Splitter and Two-Mode-Squeezing Interactions
		9.1.3 Interaction Between Qubits
		9.1.4 Nonlinear Interactions
	9.2 Atomic Ions
		9.2.1 Atom-Light Interaction
		9.2.2 Sideband Transitions: Optomechanics with an Ion
		9.2.3 Phonon–Phonon Interaction
	9.3 Superconducting Circuits
		9.3.1 Quantum-Mechanical Treatment of an LC Resonator
		9.3.2 Superconducting Quantum Bit
		9.3.3 Coupling Between a Transmon and an LC Resonator
	9.4 Optomechanical Interaction
		9.4.1 Brief Introduction
		9.4.2 Optomechanical Interaction
		9.4.3 Linearized Optomechanical Interaction
	9.5 Hybrid Quantum Systems and Cooperativity
Part IIIQuantum Information Processing and Quantum Technologies
10 Basics of Quantum Information Processing
	10.1 Quantum Gates
		10.1.1 Single-Qubit Gates
		10.1.2 Two-Qubit Gates
		10.1.3 Clifford and Non-Clifford Gates
	10.2 Quantum Circuit Model
	10.3 Measurement and Imperfections of Quantum States
		10.3.1 Projective and Generalized Measurement
		10.3.2 State Tomography
		10.3.3 Fidelity
		10.3.4 Estimation of Gate Errors
	10.4 Essential Idea of Quantum Error Correction
		10.4.1 Stabilizer Formalism
		10.4.2 Three-Qubit Repetition Code
		10.4.3 Nine-Qubit Shor Code
		10.4.4 Advanced Quantum Error Correction Codes
	10.5 DiVincenzo Criteria
11 Quantum Technologies
	11.1 Quantum Computer
		11.1.1 Grover\'s Algorithm
		11.1.2 Phase Estimation Algorithm
		11.1.3 Shor\'s Algorithm
	11.2 Quantum Key Distribution
		11.2.1 BB84
	11.3 Quantum Sensing
		11.3.1 Ramsey Interference
		11.3.2 Quantum Sensing with Entanglement
	11.4 Quantum Simulation
	11.5 Quantum Internet
A Position and Momentum Representations
Appendix B Unitary Transformation to a Rotating Frame
Appendix C Extraction of a Two-Level System from a Three-Level System
Appendix D Quantum Theory of Hydrogen Atom
	D.1  Bohr\'s Atom
	D.2  Fine Structure
	D.3  Hyperfine Structure
Appendix E Master Equation
	E.1  Density Matrix
	E.2  System-Bath Interaction
	E.3  Rate Equation
	E.4  Optical Bloch Equation
Appendix F Schrieffer–Wolff Transformation
	F.1  General Prescription
	F.2  Examples
		F.2.1  Driven Spin
		F.2.2  Generalized ``Spin\'\' Subject to the ``Driving Field\'\'
Appendix G Derivation of the SWAP4pt Gate from the Heisenberg Hamiltonian
Appendix H Cavity Cooling of a Mechanical Mode
	H.1  Quantum Noise Spectrum and Rate Equation
	H.2  Limits on the Cavity Cooling
Appendix I Entangled States and Quantum Teleportation
	I.1  Separable and Entangled States
	I.2  Measures of Entanglement
		I.2.1  Entanglement Entropy
		I.2.2  Negativity
	I.3  Quantum Teleportation
	I.4  Entanglement Swapping
	I.5  Local Operation and Classical Communication (LOCC)
Appendix J Quantum No-Cloning Theorem
Appendix  References
Index