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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