Atomic Clocks

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Atomic Clocks by Mind Map: Atomic Clocks

1. Lecture 1

1.1. History of clocks and timekeeping.

2. Lecture 2

2.1. How good is a clock?

2.1.1. Definition of SI second.

2.1.2. Clock uncertainty.

2.2. Principle of atomic clock.

2.2.1. Atomic transition provides reference frequency.

2.2.2. Advantages.

2.2.3. Interaction of atom with EM field.

2.2.4. Minimise broadening mechanisms.

2.3. How do atoms interact with EM fields?

2.3.1. Classical picture.

2.3.1.1. Electric dipole interaction.

2.3.1.2. Magnetic dipole interaction.

2.4. Electric dipole interactions.

2.4.1. Electric dipole moment.

2.4.2. Electric dipole approximation.

2.4.3. Does an atomic orbital have a dipole moment?

2.4.3.1. Expectation value of dipole moment operator.

2.4.4. The structure of hydrogen and hydrogenic atoms.

2.4.4.1. Foot §2.1 - Schroedinger equation.

2.4.4.1.1. Solution of the angular equation.

2.4.4.1.2. Solution of the radial equation.

2.4.4.2. Y&F §41.3 - Schroedinger equation for hydrogen atom.

2.4.4.2.1. Quantisation of orbital angular momentum.

2.4.4.2.2. Quantum number notation.

2.4.4.2.3. Electron probability distributions.

2.4.4.2.4. Hydrogenlike atoms.

2.4.4.3. Atkins §13.1 - The structure of hydrogenic atoms.

2.4.4.3.1. Separation of internal motion.

2.4.4.3.2. Radial solutions.

2.4.4.4. Atkins §13.2 - Atomic orbitals and their energies.

2.4.4.4.1. Energy levels.

2.4.4.4.2. Ionisation energies.

2.4.4.4.3. Shells and subshells.

2.4.4.4.4. Atomic orbitals.

2.4.4.4.5. Radial distribution functions.

2.4.4.4.6. p-orbitals.

2.4.4.4.7. d-orbitals.

2.4.5. Does a superposition of eigenstates have a dipole moment?

2.4.5.1. Electric dipole moment associated with transitions between eigenstates.

2.4.5.2. Opposite parity. Electric dipole allowed transition.

2.4.6. Foot §2.2 (up to and inc. §2.2.1) - Transitions.

2.4.6.1. Selection rules.

2.4.6.1.1. pi-transitions.

2.4.6.1.2. sigma-transitions.

2.4.7. Foot §2.2.3 - Parity.

2.5. Magnetic dipole interactions.

2.5.1. Y&F §41.4, p.1379 to 1382 - Zeeman effect.

2.5.1.1. Magnetic moment of orbiting electron.

2.5.1.2. Selection rules.

2.6. Compare electric and magnetic dipole interactions.

3. Lecture 3

3.1. Visualising electron distributions during transitions.

3.1.1. Time-dependence evolution of relative phase.

3.1.1.1. Absorption/emission of photon.

3.1.1.2. 2p0<->1s0 See movie 2p0to1s0.avi

3.1.1.2.1. Linear polarisation, pi-transition.

3.1.1.3. 2p1<->1s0 See movie 2p1to1s0.avi

3.1.1.3.1. Right circular polarisation, sigma^+ transition.

3.1.1.4. 2p-1<->1s0

3.1.1.4.1. Left circular polarisation, sigma^- transition.

3.1.1.5. 2p1<->1s0 See movie 2p1to1s0(2).avi

3.1.1.5.1. Horizontal linear polarisation, transverse observation.

3.1.2. Foot §1.8 & §1.8,1 - Zeeman effect.

3.1.2.1. pi, sigma^+ & sigma^- polarisations.

3.2. Summary of electric dipole transitions

4. Lecture 4

4.1. Spontaneous emission.

4.1.1. Explicitly forbidden by quantum mechanics. Induced by vacuum fluctuations.

4.1.1.1. Vacuum fluctuations.

4.1.2. Foot §1.7 - Einstein A & B coefficients.

4.1.3. Rate of decay. Einstein A-coefficient and lifetime.

4.1.4. Natural linewidth.

4.1.5. Calculate lifetime of H 2p state.

4.1.6. Calculate clock uncertainty of electric dipole allowed transition.

4.1.7. Is 2p<-1s in H a good clock transition?

4.2. How do we reduce Einstein A-coefficient?

4.2.1. Hyperfine splitting.

4.2.2. Einstein A-coefficient for magnetic dipole allowed transition.

4.3. Why do energy levels split?

4.3.1. Foot §2.3 (excl. §2.3.4), also §2.1.2.

4.3.1.1. Fine structure / spin-orbit splitting.

4.3.1.1.1. Quantised vector model.

4.3.1.1.2. Calculate spin-orbit splitting and energy shifts of fine structure states.

4.3.1.1.3. Selection rules.

4.3.2. Foot §4.5 - The spin-orbit interaction: a quantum mechanical approach.

4.3.3. Foot §5 & §5.1 - The LS-coupling scheme.

4.3.3.1. Fine structure in the LS-coupling scheme.

4.3.4. Y&F §41.6 - Many electron atoms and the exclusion principle.

5. Lecture 5

5.1. Why do energy levels split? Cont.

5.1.1. Proton and electron magnetic moments.

5.1.2. Electron-proton spin-spin interaction hamiltonian.

5.1.3. Origin of hyperfine splitting.

5.1.3.1. Hyperfine splitting energy from 1st order TIPT.

5.1.4. Foot §6.1 (excl. §6.1.2 & §6.1.4) - Hyperfine structure.

5.1.5. Foot §6.1.3 - Hyperfine structure for l not equal to zero.

5.1.6. Calculate spin configuration state energies of H ground state using DPT.

6. Lecture 6

6.1. Why do energy levels split? Cont.

6.1.1. Calculate spin configuration state energies of H ground state using DPT.

6.1.1.1. Calculate eigenenergies.

6.1.1.2. Calculate eigenstates.

6.2. Effect on clock uncertainty.

6.2.1. Lifetime of spontaneous emission from upper hyperfine state.

6.2.2. Is H atom clock sensitive to external magnetic fields?

6.2.3. Foot Appendix C - Magnetic dipole transitions.

6.2.3.1. Summary of electric dipole and magnetic dipole transitions.

6.3. Zeeman effect.

6.3.1. Foot § 5.5 - Zeeman effect.

6.3.1.1. Zeeman effect in LS- coupling scheme.

6.3.2. Calculate Zeeman shifts of H ground state hyperfine levels using DPT.

7. Lecture 7

7.1. Zeeman effect cont.

7.1.1. Breit-Rabi diagram of H ground state hyperfine structure..

7.1.2. Transition least sensitive to magnetic fields.

7.2. Zeeman effect and clock uncertainty.

7.2.1. H not good as a clock for practical reasons.

7.2.2. Advantages to using 'one-electron' alkali atoms for clocks, e.g. Cs.

7.2.3. Definition of SI second.

7.3. Transit time broadening.

7.3.1. Probability of being in upper state after single interaction with EM radiation (Rabi equation).

7.3.1.1. Foot §7.1, §7.3 and additional notes.

7.3.1.2. pi-pulses and pi/2-pulses.

7.3.2. Longer cavity versus slower atoms.

7.3.2.1. Foot §6.4.2 - Atomic clocks.

7.4. Light forces.

7.4.1. Foot §9.1 - The scattering force.

7.4.1.1. Momentum transfer and scattering rate.

7.4.1.2. Balance of absorption and stimulated emission.

7.4.1.3. Maximum light force.

7.4.1.3.1. Maximum light acceleration.

7.4.1.4. Atom beam stopping distance.

8. Lecture 8

8.1. Light forces cont.

8.1.1. Can a lamp be used as the photon source to stop an atom beam?

8.2. Laser cooling.

8.2.1. Doppler shift.

8.2.1.1. Laser detuning.

8.3. Laser cooling - optical molasses.

8.3.1. Foot §9.3 - The optical molasses technique.

8.3.1.1. Friction like velocity and lineshape dependent force.

8.3.1.2. Cooling rate.

8.4. Laser cooling - Doppler limit

8.4.1. Calculate Doppler limit temperature.

8.4.2. Calculate Doppler limit velocity.

8.4.3. Calculate recoil velocity.

8.4.4. Calculate capture velocity.

8.4.5. Foot §9.2 - Slowing an atomic beam.

8.4.6. Foot §9.7 - The Sisyphus cooling technique.

9. Lecture 9

9.1. Magneto-optical trap (MOT).

9.1.1. Zeeman effect.

9.1.1.1. Position dependent force.

9.1.2. Role of circularly polarised laser beams.

9.1.3. Total force on atoms.

9.2. Foot §9.4 - The magneto-optical trap.

10. Lecture 10

10.1. Caesium fountain clock.

10.1.1. Moving molasses.

10.1.2. Double pass through microwave cavity.

10.1.3. Ramsay interferometry.

10.1.3.1. Probability of being in upper state after single pass through cavity (Rabi equation).

10.1.3.2. Extra phase evolution from free flight.

10.1.3.3. Total probability of being in upper state after two passes through cavity.

10.1.3.4. Width of central Ramsay fringe.

10.1.3.5. TIPT gets it wrong, but gives good insight.

10.1.3.5.1. Foot §7.4 - Ramsay fringes.

10.1.4. Summary of caesium fountain clock.

10.2. Optical frequency standards.

10.2.1. Clock frequency instability.

10.2.2. Requirements for optical frequency standard.

10.2.2.1. Foot §8.5.3 - Optical frequency combs.

11. Lecture 11

11.1. Optical frequency standards cont.

11.1.1. Clock architecture.

11.2. Ultra-stable laser.

11.2.1. Pound-Drever-Hall (PDH) locking to ultra-stable cavity.

11.3. Linear Paul trap for ions.

11.3.1. Earnshaws theorem - saddle point, need RF.

11.3.2. Stable ion trajectories in trap.

11.4. 27Al+ ion quantum logic clock.

11.4.1. Laser cooling to quantum harmonic oscillator.

11.4.2. Elimination of first order Doppler shift in Lamb-Dicke regime.

11.4.3. Why need for the 25Mg+ ion?

11.4.3.1. Sympathetic cooling and quantum logic spectroscopy detection of clock transition in 27Al+ ion.

12. Lecture 12

12.1. 27Al+ ion quantum logic clock cont.

12.1.1. Quantum logic spectroscopy.

12.1.1.1. Pulse sequence for measuring transition frequency in 27Al+ ion.

12.2. Optical lattice clock.

12.2.1. Advantages over ion clocks.

12.3. Dipole force and optical lattices.

12.3.1. Foot §9.6 - Theory of the dipole force.

12.3.1.1. Scattering rate.

12.3.1.2. Relationship between dipole force and light shift from a.c. Stark effect.

12.3.1.2.1. Foot §7.7 - The a.c. Stark effect or light shift.

12.3.1.3. Sign of detuning determines direction of dipole force --> dipole trap.

12.3.1.4. Arrays of traps (lattices) formed from standing waves of laser light.

12.4. 87Sr optical lattice clock

12.4.1. Laser cooling transition.

12.4.2. Clock transition - detection.

12.4.2.1. Doubly forbidden --> very narrow linewidth.

12.4.3. Magic wavelength.

12.4.4. Latest research - temperature stabilised clock / lattice in cryogenic environment.

12.5. Future of atomic clocks.

12.5.1. Are fundamental constants actually constant?

12.5.2. Coupling between electromagnetism and gravity?

12.5.3. Special relativistic effects observed at very low speeds.

12.5.4. Relativistic geodesy.

12.5.4.1. Measure elasticity of Earth.

12.5.5. Nuclear clocks with 10^-19 fractional uncertainty.