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# Atomic Fountain Clock

In this course, we will use the example of the atomic clock as a tool to learn some quantum mechanics, atomic structure and atom-EM field interactions.

## Lecture 3

Learning outcomes: Know form of the time-dependent wavefunctions in transitions . Reproduce pictorially, and describe, how the electron distribution changes with time during a 2p m = 0, +/-1 - Examples class 2 & QAII 2003 & 2004 & 2005 & 2006 & 2007 & 2008. Describe how the various Delta m transitions are driven by light of differing polarisations (see also MOTs, lecture 15).

### Visualising electron distributions during transitions.

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

Dirac notation.

## Lecture 1

Learning outcomes: Know a little bit about the history of the atomic clock. 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. Describe, with the aid of a simple schematic, how an atomic clock is used to measure time? - Short question 2010 & 2011. Describe the classical picture of the electric and magnetic dipole interaction.

### How good is a clock?

Clock uncertainty.

Definition of the SI second.

### Principle of atomic clock.

Atomic transition provides reference frequency.

Interaction of atom with EM field.

### Brief history of atomic clocks

Atomic beam clocks.

Improved uncertainty.

Miniaturisation and commercialisation.

Fountain clock.

### How do atoms interact with EM fields?

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

### Lecture 2

Learning outcomes: Know how the dipole approximation leads to the form of the dipole moment operator - QAII 2003. Describe, pictorially and including some of the mathematics, how the electric dipole moment of an atom in an energy eigenstate is zero - QAII 2006. Describe how the electric dipole moment of a superposition of two states of opposite parity has a non-zero dipole moment - Weekly problem 1 & QAI 2007 & QAII 2008. Describe how non-zero electric dipole moments are associated with transitions between states with opposite parity to give selection rule. Describe the source of magnetic dipole moment, but no need to derive it. Describe how the electric interaction with the EM field is 274 times larger than the corresponding magnetic interaction - QAII 2008. Reproduce 2D representations of 3D electron distributions for s, p and d wavefunctions as well as the m = 0, +/-1 magnetic sub-state wavefunctions - Examples class 2 & QAII 2003 .

Electric dipole interactions., Electric dipole moment., Electric dipole approximation., Does an atomic orbital have a dipole moment?, Expectation value of dipole moment operator., Does a superposition of eigenstates have a dipole moment?, Electric dipole moment associated with transitions between eigenstates., Opposite parity. Electric dipole allowed transition.

Magnetic dipole interactions.

Which is stronger, electric or magnetic dipole interactions?

What happens to an atom undergoing a transition?, Visualising electron distributions in 3D., Hydrogen atom orbitals.

## Lecture 15

Learning outcomes: Describe the principles behind the atomic fountain clock and how Ramsay interferometry is used to minimise the clock uncertainty. Using TDPT, derive the probability that an atom in an initial state will be found in a different state after one pass through the microwave cavity of the Ramsay interferometer. Derive the probability after two passes through the cavity of the Ramsay interferometer. Describe what future improvements could be made to atomic clocks in the context of the things you have learned in the QAP course.

### Atomic fountain clock.

Moving molasses.

Double pass through microwave cavity.

Ramsay interferometry.

Clock uncertainty.

## Lecture 14

Learning outcomes: Describe how laser cooling works and why a velocity-dependent force leads to cooling in an optical molasses. Know the cooling rate and what the limitations of the technique are. Describe how a magneto-optical trap (MOT) works and how it can be used to overcome the limitations of using laser cooling alone. Describe the total force an atom experiences in a MOT.

### Magneto-optical trap (MOT).

Zeeman effect.

Total force on atoms.

## Lecture 4

Learning outcomes: Describe how spontaneous emission occurs when energy eigenstates are time-independent - QAII 2003. Know the Einstein A-coefficient, and its origin including how the dipole approximation is involved, and describe how this relates to the lifetime of an excited state and, therefore, the natural linewidth of the atomic transition and the atom clock uncertainty - QAII 2003. Describe how the lifetime can be calculated by evaluating the dipole matrix element - Weekly problem 1 & QAII 2005 & 2008. Describe why the 2p - QAII 2007. Describe which transitions would be good for atomic clocks, i.e. hyperfine transitions, and why - QAII 2003. Know the Einstein A-coefficient for a magnetic dipole allowed transition and why the terms in the expression make these transitions good for an atomic clock - QAII 2004. Describe the origin of the hyperfine interaction in H - QAII 2006.

### Spontaneous emission.

Explicitly forbidden by quantum mechanics., Vacuum fluctuations.

Rate of decay. Einstein A-coefficient and lifetime.

Natural linewidth.

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

### Hyperfine transitions and spin.

Hyperfine splitting., Same parity. Magnetic dipole allowed transition.

Einstein A-coefficient., Is the ground state hyperfine transition in hydrogen a good clock transition?

What causes hyperfine splitting?

How large is hyperfine splitting?

### Lecture 5

Learning outcomes: Describe the principle of time-independent perturbation theory (TIPT). Derive the first-order correction to the energy (know about Hermicity) and know how to apply it - QAI 2005 & 2006 & 2007 & 2008 & 2009. Derive the first-order correction to the wavefunction and describe why it only works for non-degenerate energy spectrum - QAI 2009.

Time-independent perturbation theory (TIPT)., First-order correction to energies., First-order correction to wavefunctions.

### Lecture 6 and lecture 7

Learning outcomes: Derive the second order correction to the energy in TIPT and know how to apply it. In time-dependent perturbation theory (TDPT), understand the derivation of the expression for the first-order probability of a transition being made upon the application of a time-dependent perturbation, but no need to be able to reproduce it. Know the expression for the probability in TDPT how to apply it - QAI 2005 & 2006 & 2007 & 2008 & 2009. Describe the time-dependence of the probability of an atom making a transition when exposed to an EM wave. Relate TDPT to atom light interactions, i.e. absorption, stimulated and spontaneous emission, and describe how spontaneous emission is a special case of stimulated emission.

TIPT second-order correction to energies.

Time-dependent perturbation theory (TDPT)., Absorption of photon., Stimulated emission., Spontaneous emission.

### Lecture 8 and lecture 9

Learning outcomes: Derive the fundamental result of degenerate perturbation theory (DPT) assuming two-fold degeneracy - QAII 2007 & 2008 & 2009. Know the expression for higher order degeneracy and know how to apply it using the example. Describe the origin of hyperfine splitting for s-states, i.e. perturbation due to penetration of proton by electron. No need for derivation, but know equation for perturbation Hamiltonian - QAII 2005 & 2008.

Degenerate perturbation theory (DPT).

Hyperfine transitions and spin cont., Spin-spin interactions., Spin-orbit interaction, Fine structure splitting., Nuclear spin - total angular momentum interaction., Hyperfine splitting., 4 degenerate spin configurations in H ground state., s-electron penetrates nucleus., First-order correction to energy.

### Lecture 10

Learning outcomes: Know how to calculate matrix elements of the  operator using Pauli spin matrices and DPT to obtain energies of spin states in H - QAII 2008. Know how to find the eigenstates of the operator using Pauli spin matrices and DPT and collate information into an energy level diagram - QAII 2004 (kind of) & QAII 2006.

Hyperfine transitions and spin cont., Use DPT to calculate energies of 4 spin configurations in H ground state., Lifetime of spontaneous emission from upper hyperfine state., Hyperfine resolved energy level diagram of H ground state.

## Lecture 11

Learning outcomes: Identify the hyperfine transition in H and show that the contribution to the clock uncertainty due to spontaneous emission for the upper hyperfine state is negligible. 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 - QAII 2003. Derive eigenvalues of spin states in H as a function of applied magnetic field - QAII 2006 & QAP 2011. Know the Breit-Rabi diagram for H - QAII 2006. Describe why m'_F = 0 QAII 2000. Describe why alkali metals are convenient for use in an atomic clock. Know the general equation for eigenvalues of atoms with any nuclear spin. Know the transit time broadening equation and how it can be a limiting factor in the minimisation of the atomic clock uncertainty and what can be done about it - QAII 2003.

### Zeeman effect.

Is H atom clock sensitive to external magnetic fields?

Use DPT to calculate energies of hyperfine states as a function of magnetic field.

Breit-Rabi diagram of H.

Transition least sensitive to magnetic fields.

H not good as a clock for practical reasons.

## Lecture 12

Learning outcomes: Describe how atoms in a resonant laser beam change momentum in the laser beam direction only. Calculate the maximum light force, acceleration and stopping distance for an atom in a resonant light beam - QAII 2001 & 2003 & 2004 & 2005 & 2006 & 2007. Calculate how many photons are required to stop a beam of atoms - QAII 2004 & 2005. Describe why it is best to use a laser as the light source as opposed to a thermal source. No need for derivation.

### Lecture 13

Learning outcomes: Describe the principle of laser action, using the rate description, and how an extended cavity diode laser can be operated to emit a single cavity mode.

Lasers., Extended cavity diode laser., Power requirements.

Laser cooling., Doppler effect., Friction like velocity dependent force., Laser detuning., Cooling rate., Limits.

### Light forces.

Momentum transfer.

Balance of absorption and stimulated emission.

Maximum light force., Maximum light acceleration., Atom beam stopping distance., Can a lamp be used as the photon source to stop an atom beam?