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Testing General Relativity
With Atom Interferometry
Part I: Atom Interferometry
James Davies
Ruth Gregory
Simon Gardiner
Low-Energy Particle Physics & Precision Measurement of Fundamental Forces
7 July 2010
Basic Configuration
• Laser-cooled atoms with 2
(+1) relevant internal states
(full/dotted lines)
• 2 counter-propagating laser
beams
Kasevich, Chu:
Appl. Phys. B 54 321 (1992)
Low-Energy Particle Physics & Precision Measurement of Fundamental Forces
7 July 2010
Effective 2-Level System
• Reduce to 1 (vertical) dimension (& assume rotating wave
approximation)
Low-Energy Particle Physics & Precision Measurement of Fundamental Forces
7 July 2010
Effective 2-Level System
• Reduce to 1 (vertical) dimension (& assume rotating wave
approximation)
Internal (electronic)
External (centre-of-mass)
Interaction
Low-Energy Particle Physics & Precision Measurement of Fundamental Forces
7 July 2010
Discrete Pulses
• Interaction is “switched” on and off
• Short, intense pulses – either the atomic evolution is “free” (no
coupling) or dominated by the interaction (internal and
external components of Hamiltonian ignored)
• π-pulses (timed to transfer atoms in state 1 to be in state 2, &
vice-versa)
• π/2-pulses (timed to transfer atoms in state 1 to be in an equal
superposition of states 1 and 2, & the reverse process)
Low-Energy Particle Physics & Precision Measurement of Fundamental Forces
7 July 2010
Pulse Sequence
• π/2 – free evolution – π –
free evolution – π/2
sequence
• Momentum kick
associated with transfer
from 1st to 2nd internal state
• No phase difference due to
internal energy difference
accrued along two “arms”
of interferometer
Kasevich, Chu:
Appl. Phys. B 54 321 (1992)
Low-Energy Particle Physics & Precision Measurement of Fundamental Forces
7 July 2010
Quantum Propagator
• Instead of operator methods, consider Feynman path
integrals
• Quantum propagator defined through
where the action is defined as
Low-Energy Particle Physics & Precision Measurement of Fundamental Forces
7 July 2010
Quadratic Lagrangian
• For a general quadratic Lagrangian
the quantum propagator may be expressed as
• Phase determined by the classical action
• Applies perturbatively for non-quadratic Lagrangians
Low-Energy Particle Physics & Precision Measurement of Fundamental Forces
7 July 2010
Spacetime Diagrams
Storey, Cohen-Tannoudji:
J. Phys. II France 4 1999 (1994)
Dimopoulos, Graham, Hogan, Kasevich:
Phys. Rev. D 78 042003 (2008)
Low-Energy Particle Physics & Precision Measurement of Fundamental Forces
7 July 2010
Particle in a (Linear) Gravitational Field
• From the Lagrangian
it is straightforward to determine classical position and
velocity as functions of time
• Hence,
Low-Energy Particle Physics & Precision Measurement of Fundamental Forces
7 July 2010
2-Level Atom Crossing a Laser Travelling Wave
Storey, Cohen-Tannoudji:
J. Phys. II France 4 1999 (1994)
Low-Energy Particle Physics & Precision Measurement of Fundamental Forces
7 July 2010
Raman Transition
Dimopoulos, Graham, Hogan, Kasevich:
Phys. Rev. D 78 042003 (2008),
Kasevich, Chu: Appl. Phys. B 54 321 (1992)
• Coupling between
internal states probably
via a Raman transition
• Counterpropagating
beams mean small
frequency difference, but
large momentum kick
Low-Energy Particle Physics & Precision Measurement of Fundamental Forces
7 July 2010
Phase Difference
• Upshot is, there is a phase difference between
interferometer “arms” given by
, yielding
observable interference fringes (counterpropagating
Raman configuration means
)
• Can determine essentially through calculating action
associated with classical trajectories
• Extend treatment to relativistic action along geodesics
[Dimopoulos, Graham, Hogan, Kasevich: Phys. Rev. D 78
042003 (2008)]
Low-Energy Particle Physics & Precision Measurement of Fundamental Forces
7 July 2010