Graduate Quantum Optics



Lecture Outline
Lecture 1  (Lectures 1 to 4 PDF)
Course Overview
Damped Harmonic Oscillator
        Steady State:  Amplitude, Phase, Quadrature Responses

Lecture 2
Frequency Modulation
        Perturbative steady state
        Frequency/Phase Modulation--Phasors
        Amplitude Modulation--Phasors
        FM vs AM in Time Domain
        FM/PM Bessel Expansion, Demo Visualization
Parametric Amplifier/Squeezing--derivation

Lecture 3
Parametric Amplifier/Squeezing--discussion
        Flow in phase space
        Quadrature Amplification   
        Classical squeezing of thermal motion
Coupled Harmonic Oscillators
         Non-degenerate, weakly coupled
             2nd order perturbation thy
             frequency repulsion
             origin of frequency repulsion due to phase response of H.O.

Lecture 4
Coupled Harmonic Oscillators (continued)
         Degenerate: Fully Mixed Modes
             Eigenfrequencies, Eigenvectors
             Example: Flopping, Normal Mode Beating, Phasor Picture
        General Solutions
            Adiabatic Approximation
            Conserved quantity--total excitation quanta
            Avoided Crossing
            DC Stark Shifts

Lecture 5  (Lectures 5 to 7 PDF)
Driven Optical Cavity
        Steady-state response
        High Finesse or Reflectivity Limit
        Phasor picture for circulating, reflected and transmitted fields
Lecture 6 
Classical Magnetic Moment in a Magnetic Field
        Precession about B
        Rotating Transverse Field: modified axial B-field
        Quantum Eq. of Motion for first order expectation values
        Geometric derivation of variation of projection Jz(t)

Lecture 7 
Classical Magnetic Moment in a Magnetic Field (cont)
        Rabi flopping & Rabi Spectroscopy
        History Lesson
Ramsey Spectroscopy
        pi/2, pi pulses etc.
        Frequency resolution in Ramsey Spectroscopy
        inhomogeneous broadening
Composite Pulse rotations        

Lecture 8 (Lectures 8 to 10 PDF)
Strategy for Understanding New Quantum System
Two-level atom dipole interaction with Laser
        Rotating Wave Approximation
        Reduction to spin 1/2 - magnetic field model
        Eigenstates, Eigenenergies
        Avoided Crossing
Lecture 9
Ensembles of Spin 1/2 objects
Angular Momentum Representation--Dicke States
Triplet-Singlet Coupling in Inhomogeneous field
Collective Operators: Jx,Jy,Jz, J+, J-
Coherent Spin State orientation noise or SQL
        (1) collective operators analysis

Lecture 10
Coherent Spin State orientation noise or SQL
        (2) Quantum Collapse/Projection Noise analysis
        (3) Interparticle-pair correlation analysis
Standard Quantum Limit and Ramsey Spectroscopy
Squeezed States:
        One-axis Twisting
            classical noise model
            development of quantum pair correlations
            experimental work so far
        Two-axis Twisting
            classical noise model
Squeezed States in Ramsey Spectroscopy

Lecture 11 (Lectures 11 to 12 PDF)
Coherent Spin States
        Minimum uncertainty/Heisenberg Relations
        Distinguishability / Husimi Q function
Summary of 1 and 2 Axis Twisting Limits
Heisenberg Limit
Squeezing Criteria
Cat or NOON States for Spectroscopy

Lecture 12
Coherent Spin States
         Angular Momentum or Dicke Basis
Generalized Cat States
Dicke State Spectroscopy
Review of Quantum Harmonic Oscillator
        Raising and Lowering Operators
Holstein-Primakoff Transformation

Lecture 13
To be posted

Lecture 14 (Lectures 14 to 15 PDF)
Husimi Q Function
        Coherent State
        Fock State
Orientation of coherent state
Time evolution of coherent state
Coherent State Displacement 
Wigner Function of Coherent State

Lecture 15
Coherent State Position Wavefunction
Coherent Cat States
Semi-classical Heterodyne and Homodyne Detection
Photo-electron current noise

Lecture 16
To be posted

Lecture 17  (Lectures 17 to 19 PDF)
Parametric Squeezing of Simple Harmonic Oscillator
        Classical equations of motion
        Hamiltonian in Schrodinger Picture
        Hamiltonian in Interaction Picture
                Resonance approximation
        Squeezing Operators
        Quadrature Operators in Heisenberg Picture
        Quadrature Noise
        Squeezed wave function
        Displacements of Squeezed Vacuum States
        Time development of Amplitude and Phase Squeezed States
Optical Analogues
Two-axis twisting analogue via Holstein-Primakoff Transformation
Lecture 18  
Quantum Beam Splitter Model
    Conservation of Energy/Quanta
    Conservation of Commutation Relations
Beam Splitter for Coherent State Displacements
Balanced Homodyne Detection

Lecture 19
Modeling Optical Loss with Beam Splitters
        Fock State Input
            Partition Noise
            Operator Description
Combining Squeezed States on 50/50 Beam Splitter
        Single Quadrature Measurements
        Joint Quadrature Measurements
        Two-mode Squeezed States
Connection to two-mode squeezing Hamiltonian
        Two-mode parametric downconversion

Lecture 20 (Lectures 20 to 24 PDF)
Mach Zehnder Interferometer
        Heisenberg Picture
        Coherent and Fock State + vacuum
Hong-Ou-Mandel Interference
Heisenberg Limited Sensitivity    

Lecture 21
 g2: 2nd Order Correlation Functions
    measurement/normal ordering   
    coherent, Fock, thermal states
    g2, Mandel Q, Fano Factor
Lecture 22
g2: Small Integration Window Limit
   Single Photon Sources vs g2<1
   Conditional single photon source
   Cauchy-Schwartz Inequality on cross-correlation
g2: Insensitivity to Optical Loss   
gn: generalization in time, space, order
Photon Anti-Bunching--single emitters
Photon Bunching--classical model/time domain
Hanbury Brown-Twiss Experiment

Lecture 23
Jaynes-Cummings Hamiltonian/ Cavity-QED
Many-atoms, low excitation limit
    coupled cavities
    normal modes
    Rabi flopping--vacuum Rabi splitting
    Avoided crossing/transmission spectrum
    Classical mass on spring model for vac. Rabi splitting
Many-atoms, Nonlinearities/Optical Bistability
Single atom-cavity Eigenstates, Eigenenergies
Photon Blockade
Cooperativity parameter

Lecture 24
Cavity-QED in closed and open boxes
How good is my box?
    Cooperativity / critical atom number
         interpretations & scalings
    Critical Photon Number
    Linewidth averaging, good vs bad cavity
Master Equation for Open Quantum Systems
    Review of Density Matrices, von Neumann Eq.
    Derivation of Master Equation for cavity
    Master Equation for two-level system
    Master Equation for cavity-QED

Lecture Demos/Links/Refs

Lecture 1

Lecture 2

Mathematica simulation of phasor at high phase modulation index

Youtube sim of parametric amplification.

Pendulum PHET simulation--see if you can parametrically amplify or de-amplify the motion.

by changing the length of the pendulum.

Lecture 3

Example of thermal noise squeezing:

Theory, Experiment

Lecture 4

Classical Rabi flopping--youtube video of two coupled pendulums

Mapping Coupled Harmonic Oscillators to Quantum 2-Level System

Lecture 5

Eric Black’s Intro. to Pound-Drever-Hall Measurements

Lecture 6

Lecture 7

Spin Echo Animation

Composite pi-pulse Animations

Lecture 8

Lecture 9

Lecture 10

1-AxisTwisting Mathematica Visualization

Thy Squeezing via Twisting and Use

Kitagawa & Ueda Twisting

Wineland_et al

Experiments on 1-Axis Twisting:

1-Axis Twisting Collisions: Left vs Right Well

1-Axis Twisting Collisions: Internal States

1-Axis Twisting Cavity-Mediated

Lecture 11

Review of Coherent Spin States

Lecture 12

Husimi Q function for Coherent Spin States

Lecture 13

Lecture 14

Lecture 15

Coherent State built up from Fock states

Lecture 16

Lecture 17

Lecture 18

Lecture 19

Lecture 20

Hong-Ou-Mandel Interference

Lecture 21

Lecture 22

Lecture 23

Lecture 24