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# Atomic Clocks

The aim of this course is to teach you about certain aspects of atomic and optical physics that, when taken as a whole, will amount to a body of knowledge that is not only relevant, but is still a very active area of research around the world. We are going to use your knowledge of quantum mechanics that you have acquired, and are still acquiring, to learn about atomic and optical physics (or more precisely – atom-light interactions) because is AOP is QM in action. Conceptually “construct” an atomic clock and use the example of such a device, and the precision measurement of time, as a learning tool.

## Lecture 1

Lecture 1, History of timekeeping lecture: There are no learning outcomes for this lecture.

## Lecture 2

Lecture 2 learning outcomes: Understand electron configurations and how to derive term symbols for atomic states. Describe what is meant by the clock uncertainty. Describe the principle of the atomic clock, including: how the atomic linewidth is related to the clock uncertainty; and, qualitatively, how the atomic clock minimises the clock uncertainty and what limits it. Explain what is meant by a "standard" in the context of metrology, in particular in the measurement of time or frequency. Weekly problem 1 Describe, with the aid of a simple schematic, how an atomic clock is used to measure time. QAP 2010 & 2011, Weekly problem 1. Describe the classical picture of the electric and magnetic dipole interaction. Know how the dipole approximation leads to the form of the dipole moment operator. QAII 2003, QAII 2008. Describe, pictorially and including some of the mathematics, how the electric dipole moment of an atom in an energy eigenstate is zero. QAII 2006, QAII 2008, QAP 2012, MAOP 2103, MAOP 2015. Foot §2.1, Y&F §41.3, Atkins §13.1 & Atkins §13.2 - Revise the solution of the Schroedinger equation for hydrogen and hydrogenlike atoms to know and understand the shape of orbitals. Describe pictorially how the electric dipole moment of a superposition of two states of opposite parity has a non-zero dipole moment. QAII 2008. Describe how non-zero electric dipole moments are associated with transitions between states with opposite parity to give selection rule. Foot §2.2 - Understand how the rate of a transition is prop. to the square of the matrix element of the perturbation. Foot §2.2.1 - Undertstand that selection rules arise from the angular integral. Foot §2.2.1 - Describe how the angular integral in a pi-transition is cylindrically symmetrical and how this leads to a Delta m = 0 selection rule. Foot §2.2.1 - Describe how the angular integral in a sigma-transition is zero unless the Delta m = +/- 1 selection rule is satisfied depending on the handedness of the circular polarisation. Foot §2.2.3 - Describe how, quantum mechanically, the origin of the parity selection rule in terms of the angular integral. Describe the source of magnetic dipole moment. QAII 2004, QAII2008. Y&F §41.4, p.1379 to 1382 - Know how the magnetic quantum number arises and how it is related to angular momentum. Understand the link with the Zeeman effect in Lectures 6/7. Describe how the electric interaction with the EM field is 274 times larger than the corresponding magnetic interaction. QAII 2008.

### How good is a clock?

Definition of SI second.

Clock uncertainty.

### Principle of atomic clock.

Atomic transition provides reference frequency.

Interaction of atom with EM field.

### How do atoms interact with EM fields?

Classical picture., Electric dipole interaction., Magnetic dipole interaction.

### Electric dipole interactions.

Electric dipole moment.

Electric dipole approximation.

Does an atomic orbital have a dipole moment?, Expectation value of dipole moment operator.

The structure of hydrogen and hydrogenic atoms., Foot §2.1 - Schroedinger equation., Solution of the angular equation., Solution of the radial equation., Y&F §41.3 - Schroedinger equation for hydrogen atom., Quantisation of orbital angular momentum., Quantum number notation., Electron probability distributions., Hydrogenlike atoms., Atkins §13.1 - The structure of hydrogenic atoms., Separation of internal motion., Radial solutions., Atkins §13.2 - Atomic orbitals and their energies., Energy levels., Ionisation energies., Shells and subshells., Atomic orbitals., Radial distribution functions., p-orbitals., d-orbitals.

Does a superposition of eigenstates have a dipole moment?, Electric dipole moment associated with transitions between eigenstates., Opposite parity. Electric dipole allowed transition.

Foot §2.2 (up to and inc. §2.2.1) - Transitions., Selection rules., pi-transitions., sigma-transitions.

Foot §2.2.3 - Parity.

### Magnetic dipole interactions.

Y&F §41.4, p.1379 to 1382 - Zeeman effect., Magnetic moment of orbiting electron., Selection rules.

## Lecture 3

Lecture 3 learning outcomes: Know that a time dependent wavefunction is proportional to the sum of the two coupled wavefunctions with the upper state wavefunction multiplied by a relative phase factor. MAOP 2015 Foot §1.8 & §1.8.1 - Understand how the pi, sigma^+ and sigma^- notation, used to describe components of emitted light in the Zeeman effect, arises and how it used when considering electric dipole transitions. Recognise link with Zeeman effect in Lectures 6/7. Reproduce pictorially, and describe, how the electron distribution changes with time during a 1s --> 2p transition in H, where Delta m = 0, +/-1. QAII 2003, QAII 2004, QAII 2005, QAII 2006, QAII 2007, QAII 2008, QAII 2009, QAP 2012, MAOP 2015, Weekly problem 2. Describe how the various Delta m transitions are driven by light of differing polarisations that depend on type of transition (e.g. pi-transition) and direction of observation. MAOP 2013.

### Visualising electron distributions during transitions.

Time-dependence evolution of relative phase., Absorption/emission of photon., 2p0<->1s0 See movie 2p0to1s0.avi, Linear polarisation, pi-transition., 2p1<->1s0 See movie 2p1to1s0.avi, Right circular polarisation, sigma^+ transition., 2p-1<->1s0, Left circular polarisation, sigma^- transition., 2p1<->1s0 See movie 2p1to1s0(2).avi, Horizontal linear polarisation, transverse observation.

Foot §1.8 & §1.8,1 - Zeeman effect., pi, sigma^+ & sigma^- polarisations.

## Lecture 4

### Spontaneous emission.

Explicitly forbidden by quantum mechanics. Induced by vacuum fluctuations., Vacuum fluctuations.

Foot §1.7 - Einstein A & B coefficients.

Rate of decay. Einstein A-coefficient and lifetime.

Natural linewidth.

Calculate lifetime of H 2p state.

Calculate clock uncertainty of electric dipole allowed transition.

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

### How do we reduce Einstein A-coefficient?

Hyperfine splitting.

Einstein A-coefficient for magnetic dipole allowed transition.

### Why do energy levels split?

Foot §2.3 (excl. §2.3.4), also §2.1.2., Fine structure / spin-orbit splitting., Quantised vector model., Calculate spin-orbit splitting and energy shifts of fine structure states., Selection rules.

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

Foot §5 & §5.1 - The LS-coupling scheme., Fine structure in the LS-coupling scheme.

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

## Lecture 5

Lecture 5 learning outcomes: Understand how spin is related to magnetic moment and particle mass. Know the spin configuratons for the H atom. QAII 2006. Know that hyperfine states arise due to coupling of nuclear spin and total electronic angular momentum (know form of HFS Hamiltonian). QAII 2005. Understand how hyperfine splitting is not due to dipole-dipole interaction. QAII 2008, QAII 2009. Describe the origin of hyperfine splitting for s-states, i.e. perturbation due to penetration of proton by electron. No need for derivation of first order energy shift. QAII 2005, QAII 2008, Weekly problem 3. Work out the possible values of F for a given atomic state. MAOP 2015 Foot §6.1 - Mostly backup reading for lecture notes, but know the expression for hyperfine structure energy in terms of F, I and J quantum numbers and recognise link with similar expression in fine structure. Derive hyperfine splitting Hamiltonian matrix for H from Pauli spin matrices. QAII 2008. Calculate the eigenvalues of the diagonal eigenstates in the hyperfine splitting Hamiltonian for H.

### Why do energy levels split? Cont.

Proton and electron magnetic moments.

Electron-proton spin-spin interaction hamiltonian.

Origin of hyperfine splitting., Hyperfine splitting energy from 1st order TIPT.

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

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

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

## Lecture 6

Lecture 6 learning outcomes: Use DPT to calculate the eigenvalues of the eigenstates coupled by off-diagonal matrix elements in the hyperfine splitting Hamiltonian for H. Know how to find which combination of spin configuration states gives rise to the eigenstates coupled by off-diagonal matrix elements in the hyperfine splitting Hamiltonian for H. Collate information on eigenstates and eigenvalues into an energy level diagram for the hyperfine spitting of the ground state of H. Know that spontaneous emission from the upper hyperfine state in the ground state of hydrogen is negligible and has a negligible contribution to clock uncertainty. MAOP 2015. Foot Appendix C - Know electric dipole and magnetic dipole selection rules from summary table. Describe how the Zeeman effect splits hyperfine states further and how, for H, the Earth’s magnetic field is significant with respect to use as an atomic clock. AL 2000. Foot §5.5 - Understand the vector model to see how the Zeeman effect arises in the LS-coupling scheme. Foot §6.3 - Understand the vector model to see how the Zeeman effect arises in the IJ-coupling scheme in the weak field, intermediate field and strong field limits. Compare to Zeeman effect in LS-coupling. Derive eigenvalues of spin states in H as a function of applied magnetic field and understand functional properties, i.e. form in low fields and in high fields. AL 2000, QAII 2003, QAII 2006, QAII 2007, QAII 2008, QAII 2009, QAP 2011, QAP 2012, MAOP 2015, Workshop 2.

### Why do energy levels split? Cont.

Calculate spin configuration state energies of H ground state using DPT., Calculate eigenenergies., Calculate eigenstates.

### Effect on clock uncertainty.

Lifetime of spontaneous emission from upper hyperfine state.

Is H atom clock sensitive to external magnetic fields?

Foot Appendix C - Magnetic dipole transitions., Summary of electric dipole and magnetic dipole transitions.

### Zeeman effect.

Foot § 5.5 - Zeeman effect., Zeeman effect in LS- coupling scheme.

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

## Lecture 7

Lecture 7 learning outcomes: Know the Breit-Rabi diagram for H. QAII 2006. Describe why m'_F = 0 AL 2000, MAOP 2015, Workshop 2. Describe why alkali metals are convenient for use in an atomic clock. Understand the functional properties of the general expression for the Zeeman effect of any one-electron atom of any nuclear spin. QAP 2012, MAOP 2015, Workshop 2. Know expression for transit-time broadened linewidth and understand its origins in the uncertainty principle. QAII 2003. Foot §7.1, §7.3 & "additional notes" is background reading to aid understanding and links to perturbation theory. No need to know derivation. Foot §7.3.1 - Know what a pi-pulse and a pi/2-pulse is in term of Rabi frequency and interaction time. Understand how transit-time broadening can be a limiting factor in the minimisation of the atomic clock uncertainty and what can be done about it. QAII 2003, MAOP 2013. Describe how atoms in a resonant laser beam change momentum in the laser beam direction only. QAII 2005, QAII 2007. Calculate the recoil velocity from emission of a photon. QAII 2003, QAII 2004, QAII 2006, QAP 2011, MAOP 2015, Workshop 3. Derive expression for maximum force. AL 2001, QAII 2007, QAP 2012. Calculate the maximum light force, acceleration and stopping distance for an atom in a resonant light beam. AL 2001, QAII 2004, QAII 2005, QAII 2006, QAII 2007, QAII 2009, QAP 2010, Weekly problem 4, Workshop 3. Calculate how many photons are required to stop a beam of atoms. QAII 2005, QAII 2007, Weekly problem 4.

### Zeeman effect cont.

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

Transition least sensitive to magnetic fields.

### Zeeman effect and clock uncertainty.

H not good as a clock for practical reasons.

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

Definition of SI second.

Probability of being in upper state after single interaction with EM radiation (Rabi equation)., Foot §7.1, §7.3 and additional notes., pi-pulses and pi/2-pulses.

Longer cavity versus slower atoms., Foot §6.4.2 - Atomic clocks.

### Light forces.

Foot §9.1 - The scattering force., Momentum transfer and scattering rate., Balance of absorption and stimulated emission., Maximum light force., Maximum light acceleration., Atom beam stopping distance.

## Lecture 8

Lecture 8 learning outcomes: Describe why it is best to use a laser as the light source as opposed to a thermal source. No need for derivation. Understand that cooling requires dissipation of energy. Understand the role of the Doppler effect and the detuning in the absorption of a photon by a moving atom. Calculate the Doppler shift per scattered photon. QAII 2007, Weekly Problem 4. Describe, using diagrams, how optical molasses in laser cooling leads to a velocity and linewidth dependent force QAP 2011, QAP 2012, Workshop 3, Workshop 4. Understand and describe how the Doppler cooling limit is arrived at (no need to derive formally). Know the expression for the Doppler cooling limit in terms of the linewidth of the cooling transition. Derive the Doppler limit velocity in terms of capture velocity and recoil velocity. Calculate the capture velocity and the recoil velocity. MAOP 2013, Workshop 3. Foot §9.2 - Slowing an atomic beam, and §9.7 - The Sisyphus cooling technique, are additional background reading (not examinable).

### Light forces cont.

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

### Laser cooling.

Doppler shift., Laser detuning.

### Laser cooling - optical molasses.

Foot §9.3 - The optical molasses technique., Friction like velocity and lineshape dependent force., Cooling rate.

### Laser cooling - Doppler limit

Calculate Doppler limit temperature.

Calculate Doppler limit velocity.

Calculate recoil velocity.

Calculate capture velocity.

Foot §9.2 - Slowing an atomic beam.

Foot §9.7 - The Sisyphus cooling technique.

## Lecture 9

Lecture 9 learning outcomes: Describe how a magneto-optical trap (MOT) works and how it can be used to overcome the limitations of using laser cooling / optical molasses alone. Describe the total force an atom experiences in a MOT. Foot §9.4 - The magneto-optical trap, has an alternative, and in my opinion, erroneous (see email to class) description of the MOT. Read for comparison purposes only.

### Magneto-optical trap (MOT).

Zeeman effect., Position dependent force.

Role of circularly polarised laser beams.

Total force on atoms.

## Lecture 10

Lecture 10 learning outcomes: Describe the principles behind the atomic fountain clock. Calculate the launch velocity in a moving molasses. Weekly Problem 3. Use Newtonian mechanics (SUVAT equations) to calculate fountain round-trip time, fountain height, etc. AL 2002, QAP 2011, MAOP 2015. Recognise equivilancy between expression for the probability of being in the upper state after a single pass through the microwave cavity and the transit-time broadened lineshape in Lecture 7. Weekly problem 5 Know and understand what a pi/2 pulse and a pi pulse is. Know that a period of free flight adds an extra complex phase evolution term to the probability coefficient. Know the expression for the width of the central Ramsay fringe. AL 2002, QAP 2011. Understand that the probability expression for the single pass and the probability expression for the double pass are exact (see additional notes, no need to know how to derive this expression). Understand how the TDPT derived expression for the probability is technically wrong and why and how its functional form is more insightful than the exact expression. Understand the terms in the clock frequency instability expression and describe how an optical frequency standard minimises the instability. Describe what is needed in an optical frequency standard. Foot §8.5.3 - Describe, briefly, the principle of an optical frequency comb.

### Caesium fountain clock.

Moving molasses.

Double pass through microwave cavity.

Ramsay interferometry., Probability of being in upper state after single pass through cavity (Rabi equation)., Extra phase evolution from free flight., Total probability of being in upper state after two passes through cavity., Width of central Ramsay fringe., TIPT gets it wrong, but gives good insight., Foot §7.4 - Ramsay fringes.

Summary of caesium fountain clock.

### Optical frequency standards.

Clock frequency instability.

Requirements for optical frequency standard., Foot §8.5.3 - Optical frequency combs.

## Lecture 11

Lecture 11 learning outcomes: Describe, with the aid of a simple schematic, how a clock based on an optical frequency standard is used to measure time and recognise the similarity with the microwave based clock schematic in Lecture 2. Describe, briefly, how the PDH technique can be used to very stably lock the clock laser (no need to know how to derive lineshape function). Describe with the aid of energy level diagrams the principle of how the 27Al+ ion quantum logic clock works in terms of why two species are needed and how the first order Doppler shift is eliminated. MAOP 2013.

### Optical frequency standards cont.

Clock architecture.

### Ultra-stable laser.

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

### Linear Paul trap for ions.

Earnshaws theorem - saddle point, need RF.

Stable ion trajectories in trap.

### 27Al+ ion quantum logic clock.

Laser cooling to quantum harmonic oscillator.

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

Why need for the 25Mg+ ion?, Sympathetic cooling and quantum logic spectroscopy detection of clock transition in 27Al+ ion.

## Lecture 12

Lecture 12 learning outcomes: Describe how quantum logic spectroscopy works in the 27Al+ clock. No need to memorise the whole of Slide 1, but it may aid your description if you know the pulse sequence and understand the effect of each step in the sequence. Describe the relative advantages of the 27Al+ quantum logic clock, fountain clocks and optical lattice clocks with respect to each other. MAOP 2015 Foot §7.7 - Recognise DPT in the formulation of the a.c. Stark effect. Foot §7.7 -Describe, with the aid of an energy level diagram, the a.c. Stark shift as a function of Rabi frequency for negative and positive detunings. Describe how the dipole force is the negative gradient of the potential and how this is related to the light shift from the a.c. Stark effect. Know the sign of the dipole force with respect to the sign of the laser detuning and therefore describe how optical dipole traps and optical lattices trap atoms. Foot §9.6 (Dipole trapping of sodium atoms) - Understand how a focused (Gaussian) laser beam has a finite sized waist where the light is most intense in the middle of the waist (the origin). MAOP 2013, Workshop 5. Calculate the depth of an optical dipole trap / optical lattice site. MAOP 2013, Workshop 5. Describe with the aid of an energy level diagram how the differential Zeeman shift is eliminated in the 87Sr optical lattice clock. Describe the principle of the magic wavelength with respect to the 87Sr optical lattice clock. Describe in terms of selection rules why the clock transitions in optical frequency standards are so narrow. MAOP 2013. Describe what limits the performance of the 87Sr optical lattice clock.

### 27Al+ ion quantum logic clock cont.

Quantum logic spectroscopy., Pulse sequence for measuring transition frequency in 27Al+ ion.

### Dipole force and optical lattices.

Foot §9.6 - Theory of the dipole force., Scattering rate., Relationship between dipole force and light shift from a.c. Stark effect., Foot §7.7 - The a.c. Stark effect or light shift., Sign of detuning determines direction of dipole force --> dipole trap., Arrays of traps (lattices) formed from standing waves of laser light.

### 87Sr optical lattice clock

Laser cooling transition.

Clock transition - detection., Doubly forbidden --> very narrow linewidth.

Magic wavelength.

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

### Future of atomic clocks.

Are fundamental constants actually constant?

Coupling between electromagnetism and gravity?

Special relativistic effects observed at very low speeds.

Relativistic geodesy., Measure elasticity of Earth.

Nuclear clocks with 10^-19 fractional uncertainty.