Atomic and Optical Physics II

What this course covers

• Defining AMO Physics
• Evolution from Individual Particles to Many-Body Physics
• AMO Physics as Low-Energy Quantum Physics
• Three Paradigmatic Hamiltonians
• Historical Development of AMO Physics
• Control Over Quantum Systems
• Atoms in an Optical Lattice
• Entropy, Control, and Complexity in Quantum Systems
• Technology Drivers of AMO Advances
• Longitudinal vs. transverse field separation
• Spatial Fourier decomposition of Maxwell's equations
• Independent degrees of freedom of the electromagnetic field
• Coulomb gauge and the vector potential
• Electromagnetic energy as a sum of harmonic oscillators
• Normal mode coordinates of the radiation field
• Quantization of the electromagnetic field
• Full atom-field Hamiltonian from first principles
• Rigorous derivation of the electric dipole approximation
• Course roadmap: atoms, photons, interactions
• Photons as harmonic oscillators
• Unit overview: six stages of quantum light
• Single-mode electromagnetic field Hamiltonian
• Position and momentum operators of the field mode
• Electric field operator as momentum quadrature
• Heisenberg vs. Schrödinger picture for field operators
• Fock (number) states of the electromagnetic field
• Coherent state as eigenstate of annihilation operator
• Quasi-probability distributions (Q, P, W)
• Phase-space portraits of key quantum states
• Time evolution as rotation in phase space
• Coherent states as minimum-uncertainty states
• Wigner distribution and negative quasi-probabilities
• Second-order coherence function G²(τ)
• Fano factor and photon-number statistics
• Quantum states of light overview
• Squeezed states: concept and motivation
• Classical squeezing via parametric drive
• Slowly varying amplitude (rotating wave) approximation
• Quadrature amplification and de-amplification equations
• Optical parametric down-conversion
• Squeezing Hamiltonian and operator
• Strong-pump (classical field) approximation
• Action of squeezing operator on quadratures
• Squeezed vacuum state
• Phase space representation of a single mode
• Quadrature operators A1 and A2
• Uncertainty relation between quadratures
• Coherent state as the quantum classical limit
• Quadrature squeezing
• Classical vs. quantum squeezing
• Beam splitter as a mode mixer
• Balanced homodyne detection
• Measuring quadrature noise with homodyne detection
• Shot noise: measurement vs. field property
• Coherent states and minimum uncertainty
• Signatures of non-classical light
• Photon number and coherent-state current
• Beam splitter splitting of coherent states
• Vacuum port and noise addition at a beam splitter
• Sum and difference of split-beam photo-currents
• Squeezed vacuum injection and noise reduction
• Dual-rail photon qubit representation
• Phase shifter as a quantum operation
• Beam splitter Hamiltonian and matrix representation
• Beam splitter action on single-photon states
• Dual-rail Hilbert space and qubit states
• Beam splitter as Y-axis rotation on Bloch sphere
• Phase shifter as Z-axis rotation on Bloch sphere
• Universal single-qubit gates via Bloch's theorem
• Mach-Zehnder interferometer as a quantum gate
• Nonlinear (Kerr) medium and cross-phase modulation
• Two-qubit gate via nonlinear Mach-Zehnder interferometer
• Definition of quantum entanglement
• EPR singlet state and Bell states
• Entanglement requires separable subsystems
• Polarization-entangled photon pairs
• Non-local correlations and fragility of entangled states
• Entanglement purification protocol
• Partial trace and reduced density matrix
• Von Neumann entropy as a measure of entanglement
• Single-photon entanglement ambiguity
• Density matrix as partial trace of a pure state
• Hong-Ou-Mandel interference
• Entangling remote atoms via photon detection
• Bell states as measurement outcomes
• CHSH Bell inequality
• Closing the detection loophole with atomic qubits
• Standard quantum (shot noise) limit in interferometry
• Heisenberg limit via entangled (NOON-state) photons
• NV centers as solid-state atom traps
• Shot noise limit in interferometry
• Heisenberg limit for phase measurement
• Entangled-state interferometer using quantum gates
• NOON states
• Fragility of entangled states to particle loss
• Squeezed light interferometry
• Super beam splitter and BEC implementation
• Classical probability for independent particles
• Quantum indistinguishability and counting statistics
• G2 correlation function
• Bosonic bunching: G2=2 for thermal light and thermal bosons
• G2=1 for Bose-Einstein condensates and lasers
• Impact of G2 on two-body and three-body collision rates
• Wave interference as origin of intensity fluctuations
• Quantum operator approach: exchange terms for bosons and fermions
• Hanbury Brown-Twiss effect and experimental measurement of G2
• Single-photon Kerr effect
• G2 function and single-mode light reconciliation
• Thermal state statistical operator
• Feynman diagrams as exact perturbative terms
• Time evolution operator and Dyson series
• Energy conservation and the S and T matrices
• Fermi's golden rule from perturbation theory
• Virtual states and energy–time uncertainty
• Physical reality of virtual processes
• Classical failure to explain neutral atom attraction
• Quantum zero-point fluctuations of atomic dipole moments
• Van der Waals potential via perturbation theory
• Coupled LC oscillator analogy for van der Waals force
• Correlated atomic dipole fluctuation mechanism
• Vacuum field fluctuations as an alternative mechanism
• Complementarity of zero-point energy descriptions
• Van der Waals interaction between neutral atoms
• Diagrammatic origin of retardation in Van der Waals forces
• Polarizability of a conducting sphere
• Atom–wall Casimir-Polder potential
• Casimir force between two metal plates: scaling argument
• Mode structure between parallel metal plates
• Density of states for cavity modes
• Zero-point energy summation and UV divergence
• Regularization and the Casimir potential derivation
• Reality of zero-point energy and dark energy
• Role of the fine structure constant in Casimir physics
• Resonance divergence in perturbation theory
• Fourier/Laplace transform of the time evolution operator
• Iterative expansion of the resolvent as a series
• Matrix elements of G₀ in the unperturbed basis
• Diagrammatic representation of the resolvent series
• The kernel R_B(Z) — regrouping resonant terms
• Exact geometric summation of G_BB
• Divergence-free approximation via the kernel
• Open quantum systems and master equations
• Density matrix formalism
• Hamiltonian (unitary) evolution of the density matrix
• Beam splitter as a measurement model
• Measurement back-action and wave function update
• Deriving the differential equation for the density matrix
• Born approximation for the environment
• Markov approximation and memory-free environments
• Separation of time scales and coarse-grained evolution
• Partial trace over the environment
• Lindblad / Liouvillian master equation structure
• Born and Markov Approximations in Master Equations
• Violations of Born and Markov Approximations
• Spectrum of Light Emitted at Low Intensity
• Driven Atom as a Harmonic Oscillator
• Mollow Triplet at High Intensity
• Lindblad Master Equation and Jump Operators
• Bloch Vector and Optical Bloch Equations
• Mollow Triplet and Dressed States
• Linewidths of Mollow Triplet via Eigenvalues
• Steady-State Solution of the Bloch Equations
• Absorption and the In-Quadrature Bloch Component
• Equality of Absorbed and Emitted Photon Rates in Steady State
• Power Broadening and Saturation
• Saturation Parameter and Population Inversion Limit
• Single quantum systems and quantum jumps
• Ensemble average vs. single-system behavior
• Non-uniqueness of density matrix unraveling
• Quantum Monte Carlo wave function method
• Wave function change by a zero (null) measurement
• Non-Hermitian effective Hamiltonian for no-jump evolution
• Equivalence of quantum trajectories to the master equation
• Fully quantized dipole Hamiltonian
• Coherent state to classical field transformation
• Optical Bloch equations and the Bloch vector
• In-phase and in-quadrature dipole moment components
• Optical force as gradient of the Hamiltonian
• Reactive (dipole) force and dissipative (spontaneous) force
• Traveling wave vs. standing wave force geometry
• Spontaneous scattering force and photon momentum transfer
• Adiabatic elimination of internal atomic dynamics
• Spontaneous light force and scattering rate
• Optical molasses and viscous damping
• Doppler cooling limit
• Atomic beam slowing with a single laser
• Chirped laser slowing technique
• Decelerating reference frame and fictitious force
• Stable lock point in velocity space
• Exact Doppler limit for single-beam slowing
• Velocity distribution sweep in chirped slowing
• Dressed atom picture basics
• Dressed state diagonalization and mixing angle
• Rate equations and dressed state populations
• Mollow triplet structure and linewidths
• Radiative cascade: bare vs. dressed basis pictures
• Dressed energy levels in a spatially varying laser field
• Dipole force as gradient of dressed level energies
• Force fluctuations and heating at resonance
• Velocity-dependent force and sign reversal of friction
• Electric vs. Lorentz force in optical traps
• Energy conservation in Sisyphus cooling
• Energy conservation in conservative dipole traps
• Wing's theorem and weak-field seeking states
• One-dimensional magnetic trapping demonstration
• Saddle points and 3D trap stability
• Gyroscopic stabilisation in magnetic trapping
• The Levitron as a magnetic trap analogue
• Effect of spin rate on trap stability (Majorana analogy)
• Larmor precession and adiabatic following
• G-factor as the classical spin-rate analogue
• Evaporative cooling as the route to quantum degeneracy
• Bogoliubov approximation for interacting Bose gas
• Bogoliubov transformation and quasi-particle dispersion
• Measuring speed of sound in a BEC
• Quantum depletion of the condensate
• Dynamic structure factor and quasi-particle spectroscopy
• Gross-Pitaevskii equation for inhomogeneous BEC
• Thomas-Fermi approximation and condensate density profile
• Healing length
• Vortices in BECs and the GP equation
• Optical lattices and the tight-binding limit
• Fermionic vs. Bosonic cooling behavior
• Fermionic pairing and superfluidity
• Feshbach resonance
• BEC-BCS crossover
• Many-body vs. two-body nature of Cooper pairing
• Bound states and dimensionality
• Boundary conditions in radial Schrödinger equation
• Density of states and bound-state condition in Fourier space
• Ions vs. neutral atoms: trapping forces
• Sideband cooling of trapped ions
• Scientific applications of ion traps
• Earnshaw's theorem and dynamic trapping
• Penning trap operation
• Paul (RF) trap: alternating confinement principle
• Micro-motion and the trapping potential
• Single-qubit operations in ion traps
• CNOT gate and universal quantum computation

Syllabus

1. Lectures

   1. 1. Introduction to Atomic Physics
   2. 2. QED Hamiltonian
   3. 3. Quantum description of light, Part 1
   4. 3. Quantum description of light, Part 2
   5. 4. Non-classical light, squeezing, Part 1
   6. 4. Non-classical light, squeezing, Part 2
   7. 5. Single photons, Part 1
   8. 5. Single photons, Part 2
   9. 6. Entangled states
   10. 7. Metrology, shot noise and Heisenberg limit, Part 1
   11. 7. Metrology, shot noise and Heisenberg limit, Part 2
   12. 8. g(2) for atoms and light
   13. 9. Diagrams for light-atom interactions
   14. 10. van der Waals and Casimir interactions
   15. 11. Casimir force
   16. 12. Resonant interactions
   17. 13. Derivation of optical Bloch equations
   18. 14. Solutions of optical Bloch equations, Part 1
   19. 14. Solutions of optical Bloch equations, Part 2
   20. 15. Unraveling Open System Quantum Dynamics
   21. 16. Light forces, Part 1
   22. 16. Light forces, Part 2
   23. 17. Dressed atom, Part 1
   24. 17. Dressed atom, Part 2
   25. 18. Techniques for ultralow temperatures
   26. 19. Bose gases
   27. 20. Fermi gases, BEC-BCS crossover
   28. 21. Ion trapping and quantum gates