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Atomic Fountain Clock
by David Carty
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Atomic Fountain Clock

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Lecture 3

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Visualising electron distributions during transitions.

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Lecture 1

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How good is a clock?

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Principle of atomic clock.

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Brief history of atomic clocks

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How do atoms interact with EM fields?

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Lecture 2

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Lecture 15

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Atomic fountain clock.

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Future of atomic clocks.

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Lecture 14

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Magneto-optical trap (MOT).

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Lecture 4

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Spontaneous emission.

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Hyperfine transitions and spin.

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Lecture 5

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Lecture 6 and lecture 7

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Lecture 8 and lecture 9

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Lecture 10

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Lecture 11

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Is the ground state hyperfine transition in H suitable for an atomic clock?

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Zeeman effect.

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

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Transit time broadening.

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Lecture 12

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Lecture 13

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Light forces.

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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.

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).

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.

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 .

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.