The Quantum World of Ultra-Cold Atoms and Light Book II: The Physics of Quantum-Optical Devices
The Table of Contents for the book is as follows:
I PRINCIPLES OF QUANTUM DEVICES
The Science and Technology of Quantum Optics
Quantum Mechanics—from Paradox to Paradigm
Coherence and Quantum Optics
Quantum Optics—its Scope and its Aims
Preparation, Control and Measurement of Composite Quantum Systems
Quantum Devices
Quantum Information Theory
Theoretical Quantum Optics
The Quantum Optical System-Environment Paradigm
The Electromagnetic Field as an Environment
Inputs and Outputs
Quantum Control and Quantum Processing
Implementing Quantum Control Using Trapped Ions
Several Trapped Ions—the Quantum Processor
Distributing Quantum Information
Quantum Networks
Cavity QED Systems
Superconducting Systems
Technologies for Quantum Processing
Summary
From Atoms to Qubits
Representing and Processing Quantum Information
Linear Unitary Evolution
Quantum Parallelism and Projective Measurements
The DiVincenzo Criteria for a Practical Quantum Computer
Universal Sets of Quantum Gates
One-Qubit Gates
Two-Qubit Gates
Two Different Universal Sets of Gates
Quantum Algorithms
Efficient Arithmetic Operations on a Quantum Computer
Determination of Periodicities Using Quantum Measurements
The Universal Quantum Simulator
Efficient Quantum Simulation
The Tasks for a Quantum Simulator
Quantum Information
Entanglement
Entanglement of Pure States
Bipartite Entangled States
Bipartite Entanglement of Qubits
The Schmidt Decomposition
Multipartite Entanglement
Decoherence, Mixed States and Entanglement
Transferring Quantum Information
The No-Cloning Theorem
Teleportation
II COHERENT OPTICAL MANIPULATION OF ATOMS
The Two-Level System Approximation
An Atom Driven by a Classical Driving Field
Schrödinger Equation for a Multilevel Atom
Non-Resonant Driving and the AC Stark Shift
Two-Level Atoms and the AC Stark Shift
Coherent Manipulation of Two-Level Systems
The Ideal Two-Level Atom Driven by a Classical Field
General Pulse Shape
Validity of the Rotating Wave Approximation
The Rabi Problem for a Square Pulse
The Pseudospin Formalism
The Bloch Sphere
The Bloch Equation
Dressed States of the Rabi Hamiltonian
Eigenvectors—the Dressed States
Eigenvalues—the Dressed Energy Levels
Quantum State Engineering
Adiabatic Passage along Dressed States
The Adiabatic Theorem and Berry's Phase
Coherent Manipulation of Multilevel Systems
Three-Level Systems
The Λ-Configuration
Experimental Realization of the Λ-Configuration
Far-Detuned Raman Processes
The Dark State Configuration
Approximations in the Far-Detuned Case
Effect of Spontaneous Emission
Quantum State Engineering in the Raman Configuration
Adiabatic Population Transfer along a Dark State—STIRAP
Conditions for the Validity of the Adiabatic Transfer Procedure
The Two-Level System Including Atomic Motion
Hamiltonian Including Kinetic and Electronic Terms
Interaction with a Travelling Wave
Basis States
Equations of Motion
Motion of an Atom in a Standing Light Wave
Basis States
Equations of Motion
Off-Resonant Excitation—Optical Lattice Potentials
Momentum Transfer by Adiabatic Passage
Solutions of the Equations of Motion
Modelling Real Atoms
The Alkali Atoms
Separation of Valence Electron and Core Electrons
Electrostatic Interactions
The Radial Wavefunction of the Valence Electron
Atoms with More than One Valence Electron
Spectroscopic Notation
L-S Coupling of Angular Momenta
Spectral Terms for Alkali Atoms
Fine and Hyperfine Structure
Magnetic Moments
Electric Quadrupole Moment of the Nucleus
Spin Orbit Interaction and Fine Structure
Nuclear Spin and Hyperfine Structure
The Zeeman Effect
Interaction with Electromagnetic Radiation
Centre of Mass Hamiltonian
Electric Dipole Radiation for a Multilevel Transition
Electric Dipole Matrix Elements and Rabi Frequency
Spontaneous Emission Rate for a Jg → Je Transition
Selection Rules, Polarization and Angular Distribution
Total Transition Rate
Example
Fine Structure and Hyperfine Structure
Appendix: Spherical Tensors and the Wigner–Eckart Theorem
III ATOMS AND QUANTIZED OPTICAL FIELDS
The Quantum Stochastic Schrödinger Equation
The Quantum Optical System-Environment Model
The Hamiltonian
The Coupling Operators
Examples of Hamiltonians and Coupling Operators
Physics of the Quantum Stochastic Schrödinger Equation
The Schrödinger Equation in the Interaction Picture
The Noise Operator
The Born–Markov Approximation
Line-Shifts and Renormalization
Formulation of Quantum Stochastic Calculus
Coarse Graining in Time and the Quantum Ito Increments
The Quantum Noise Hilbert Space
Solution of the Schrödinger Equation for a Finite Time Interval
Quantum Stochastic Schrödinger Equation in the Interaction Picture
Transformation back to the Schrödinger Picture
Ito Stochastic Increments in the Schrödinger Picture
Quantum Stochastic Schrödinger Equation in the Schrödinger Picture
The Central Role of the Quantum Stochastic Schrödinger Equation
Solving the Quantum Stochastic Schrödinger Equation
The Evolution Operator
Coupling to Several Independent Light Fields
Inclusion of a Coherent Input Field
Quantum Stochastic Schrödinger Equation
Interaction with a Finite Temperature Light Field
Finite Temperature Ito Increments
Ensemble Formulation for Finite Temperature Fields
Derivation of the Quantum Stochastic Differential Equation
Evaluation of Thermal Averages
Finite Temperature Evolution Operator
The Master Equation
The Master Equation within the Quantum Stochastic Framework
Derivation of the Master Equation
Superoperator Notations for Master Equation Operators
Non-Lindblad Master Equations
Time Correlation Functions
The Evolution Operator for the Master Equation
Relation to the Full Evolution Operator
Multitime Averages
Quantum Regression Theorem
Applications of the Master Equation
Inputs, Outputs and Quantum Langevin Equations
The Quantum Langevin Equation
The Quantum Langevin Equation
Noise Terms
The Quantum Langevin Equation in Terms of Outputs
Relationship between Input and Output
Time-Reversed Quantum Langevin Equation
Inputs and Outputs, and Causality
Applications of the Output-Driven Quantum Langevin Equation
Correlation Functions of the Output
Output Correlations Functions and System Correlation Functions
Quantum Stochastic Differential Equations
Development of the Quantum Stochastic Differential Equation
Alternative Form
Correspondence with the Quantum Langevin Equation
Phase Shift on Reflection From a Cavity
Several Inputs and Outputs
Fermionic Input-Output Theory
Cascaded Quantum Systems
Coupling Equations
Interpretation of the Quantum Langevin Equation
Quantum Stochastic Formalism
Quantum Stochastic Differential Equation
Master Equation
Quantum Stochastic Schrödinger Equation
Applications
Driving a Quantum System with Light of Arbitrary Statistics
Cascaded Quantum Networks
Dissipative Dynamics of Driven Atoms
Formulation of the Optical Bloch Equations for a Two-Level System
System Hamiltonian
The Master Equation
Explicit Form and solutions of the Optical Bloch Equations
The Zero Driving Field Case
Stationary Solution in the Presence of a Coherent Driving Field
Electric Polarization and Susceptibility
Time-Dependent Solutions
Dissipative Dynamics of the Λ-System
Parameters for the Three-Level System
System Hamiltonian
Master Equation
Multi-Atom Optical Bloch Equations
Coupling to Several Non-Independent Light Fields
Hamiltonian and Noise Operators
Decay Constants and Energy Shifts
Evaluation of Damping Constants
Evaluation of Lineshifts
Formulation of the Quantum Stochastic Equations
Correlated Quantum Ito Increments
Quantum Stochastic Differential Equation
The Many Atom Master Equation
Spontaneous Emission Rates and Dipole-Dipole Forces
Special Cases
Superradiance
Appendix: Spherical Bessel Functions
IV LASER COOLING
Quantum Stochastic Equations for Laser Cooling of Atoms
Notation and Hamiltonians
Electromagnetic Field and Hamiltonian
System Hamiltonian
Interaction Hamiltonian
Quantum Stochastic Differential Equation Formalism
Noise Operators
Total Noise Operator
Density of States
Quantum Noise Increments
Quantum Stochastic Schrödinger Equation
Equivalence to a Single Set of Noise Increments
Recoil Effects
The Model
The Atom
The Radiation Field
The Interaction Hamiltonian
Quantum Stochastic Schrödinger Equation Including Atomic Motion
The Laser Cooling Master Equation
Laser Cooling of Untrapped Atoms
Wigner Representation of the Atomic Density Matrix
Internal and Centre of Mass Degrees of Freedom
Evolution Equation for the Wigner Distribution Function
Doppler Cooling of a Two-Level System Using a Travelling Light Wave
Equation for the Wigner Function
Evolution Operators and Stationary Solutions
Projectors
Adiabatic Elimination Procedure
The Laser Cooling Fokker–Planck Equation
Evaluation of the Diffusion Coefficient Dzz
Summary
Laser Cooling of a Trapped Ion
Formulation of a One-Dimensional Model
Hamiltonian Terms
Laser Cooling Terms
The Full Master Equation for Cooling of a Trapped Ion
Perturbation in Terms of the Lamb–Dicke Parameter
Perturbative Expansion of the Master Equation
Elimination of the Internal Degrees of Freedom
Analysis of the Ion Trap Master Equation
The Harmonic Oscillator Trap
Solutions of the Ion Trap Master Equation
Analysis of the Process of Cooling
Doppler Cooling of a Trapped Ion
Sideband Cooling
V CONTINUOUS MEASUREMENT AND QUANTUM TRAJECTORIES
Continuous Measurement
Photon Counting
The Normally Ordered Counting Formulae
Measurement Operators for the Electromagnetic Field
Photon Counting Formulae
Photon Counting Statistics
Classical and Non-Classical Light
Photon Counting Correlation Functions
Quantum Operations
Definition of a Quantum Operation
Quantum Operations on the Wavefunction
Measurement Operators for the Electromagnetic Field
Photon Counting Using the Quantum Stochastic Schrödinger Equation
Compact Notation
Formulation of Photon Counting Using Quantum Operations
Quantum Operations Induced on the System
Probability Density of the First Detection Time
Resolution in Terms of Photon Counts
Photon Count Hierarchies
Resolution of the System Density Operator
Comparison with the Classical Poisson Process
Unravelling the System Density Operator
Photon Counting Statistics
Density Operator after n Counts
Quantum Trajectories
Stochastic Wavefunction Simulations
Stochastic Interpretation
Hierarchy of Equations
Probabilities
Stochastic Wavefunction Algorithms
Applications
Spontaneous Emission from a Two-Level Atom
The Driven Two-Level System
The Damped Cavity Mode
Quantum Jumps in Three-Level Systems
Theoretical Description
Photon Detections in Terms of the Delay Function
VI PHASE-SENSITIVE QUANTUM OPTICS
Homodyne Measurement
Procedure for Homodyne and Heterodyne Detection
General Formulae
Coherent Signal Detection
Balanced Homodyne/Heterodyne Detection
Appendix: Quantum Stochastic Equations for Homodyne Measurement
Ideal Homodyne Measurement
Quadrature Phases and Homodyne Current
Ito Increments for Quadrature Phases
Measurement Operators for Eigenvalues with a Continuous Range
Formulation of the Quantum Stochastic Schrödinger Equation
Master Equation
Continuous Measurement of the Quadrature Phase Components
Squeezing, Quantum Correlations and Quantum Amplifiers
Single-Mode Squeezing and Quantum Noise Reduction
Heisenberg's Uncertainty Principle
Quadrature Phases
Defining Squeezing
Squeezed States of the Harmonic Oscillator
Definition of an Ideal Squeezed State
Production and Measurement of Squeezed Light
The Degenerate Parametric Amplifier
Quantum Langevin Equation
Squeezing Produced
Input-Output View of Squeezing and the Ideal Phase-Sensitive Amplifier
Two-Mode Squeezing and Correlated Quanta
Quadrature Phases
Quantum Limited Phase-Insensitive Amplifiers
Identical Couplings to the Input and Output
Input and Output Coupled to Only One Mode
Quantum Limits on Noise in Linear Amplifiers
Fundamental Limits on Added Quantum Noise—the Caves Amplifier Noise Bound
Attenuators and Beam Splitters
VII QUANTUM PROCESSING WITH ATOMS, PHOTONS AND PHONONS
Cavity Quantum Electrodynamics
The Jaynes–Cummings Model
Implementations of Cavity Quantum Electrodynamics
Details of the Jaynes–Cummings Hamiltonian
Energy Levels and Eigenstates
Eigenvalues and Eigenvectors—the Dressed States
The Dressed Hamiltonian in the Far-Detuned Limit
Rabi Oscillations in the Jaynes–Cummings Model
Interaction with the Environment
Master Equation
The Strong Coupling Condition
Optical Manipulation of Trapped Ions
The Trapped Ion Hamiltonian
Components of the Trapped Ion Hamiltonian
Electronic excitation
Interaction with Laser Light
Manipulation of the Quantum State of the Centre of Mass
The Lamb–Dicke Regime
Total Hamiltonian in the Rotating Frame
Energy Spectrum and Eigenstates of the Bare Hamiltonian
Interaction with the Quantized Electromagnetic Field
Laser-Induced Couplings in the Lamb–Dicke Regime
Approximate Forms in the Lamb–Dicke Regime
Effective Hamiltonians Arising from Laser Induced Couplings
Quantum State Engineering
Superpositions of Electronic States
Conversion of Electronic Superpositions to Motional Superpositions
Generation of an Arbitrary Superposition of Motional States
Appendix: Radio-Frequency Ion Traps
Classical Equations of Motion
Quantum Theory of Ion Traps
The Ion Trap Quantum Computer
Ions in a Linear Trap
Two Ions in a Linear Trap
N Atoms in a Linear Trap
Implementation of a Two-Qubit Quantum Gate
Representation of Qubits
Manipulation of Qubits
Basic Gate Operations
The Controlled Phase Gate
Mølmer–Sørensen Gate
Gate Hamiltonian
Transition Paths
Effective Spin Hamiltonian
Use as a Two-Qubit Gate
Creation of GHZ-Like States
Creation of Many-Body Entanglement
Summary
Geometric Phase Gates
Geometric Phase in a Harmonic Oscillator
Phase of Two Ions
The Ion Trap Quantum Computer in Practice
Quantum Computing Using Multiple Ion Traps on a Chip
VIII CIRCUIT QUANTUM ELECTRODYNAMICS
Quantum Circuit Theory
The LC Oscillator
Lagrangian and Hamiltonian
Quantization of the Oscillator
The Transmission Line
Transmission Line Wave Equation
The Flux Potential and the Lagrangian Formulation
Boundary Conditions
Transmission Lines Coupled to Circuits
Parallel Coupling of the LC Oscillator to a Transmission Line
Series Coupling of the LC Oscillator to a Transmission Line
An Oscillator Embedded in a Transmission Line
Lagrangian Formulation of Circuits Coupled to Transmission Lines
Voltage Coupling
Quantization of the Transmission Line
The Finite Transmission Line
Coupling to an LC Circuit
Appendix: The Wave Equation in One Dimension
Green's Functions
The Causal Green's Function
Time-Dependent Source of Fixed Shape
Superconducting Quantum Devices
The Josephson junction
Analysis of the Hamiltonian
Flux and Charge as Canonical Co-ordinate and Momentum
Operator Form of the Hamiltonian
Equations of Motion for a Josephson Junction
The Josephson Junction as a Non-Linear Inductor
The Josephson Junction as a Circuit Element
Josephson Oscillations
The Open Circuit Configuration
Qubit Architectures
The Transmon Qubit
Formulation as Cavity QED
Measurement of the Output
Other Qubit Architectures
Josephson Junction Amplification
The Josephson Ring Modulator
IX INTERFACING QUANTUM NETWORKS
Cavity Quantum Electrodynamics Networks
The Cavity QED Quantum Memory
Hamiltonian for the Atom-Cavity System
Adiabatic Elimination of the Excited State
Interaction with Input and Output Fields
Quantum Information Transfer to the Electromagnetic Field
A Programmable Single Photon Source
Quantum Information Transfer between Nodes
The Problem of Photon Reflection from the Second Cavity
The Cavity QED Model of Quantum Information Transmission
Achieving Ideal Quantum Transmission
Use of the Quantum Trajectory Picture
The Dark-State Ensemble Quantum Memory
Quantum Memory Using an Ensemble of Λ-Systems
Hamiltonian for N Three-Level Atoms
The Family of Dark States
The Principle of the Dark State Quantum Memory
Approximate Equations of Motion
The Harmonic Approximation to an Ensemble of Atoms
Equations of Motion in the Harmonic Oscillator Approximation
Quantum State Transfer and Quantum Memory
The Λ-I Configuration
The Information Transfer Process
Storing the Quantum Information
Transferring the Stored Input to the Output
The Λ-II Configuration
Spatially Extended Atomic Ensembles
Light Propagation in an Atomic Vapour—Semiclassical Theory
Electric Polarization and Susceptibility
Perturbative Regime in the Probe Field—Susceptibility
Electromagnetically Induced Transparency
Propagation in One Dimension
The Wave Equation
Absorptive and Dispersive Behaviour
The Transparency Window
Slow Light
Quantum Theory of Light Propagation in an Atomic Vapour
One-Dimensional Electromagnetic Field Operators
Spatially Dependent Collective Atomic Operators
Hamiltonian and Equations of Motion
Solutions of the Equations of Motion
Quantum Memory Using Dark State Polaritons
Interpretation as Dark-State Polaritons
Stopping and Re-Accelerating Photon Wavepackets
References
Author Index
Subject Index
