MATTER- WAVE INTERFER OMETER S" A SYNTHETIC APPROACH

For example, a rectangular potential profile in space is obtained ...... in a given internal state is considered an elementary particle with a mass corre- .... the tensor-field theory of gravity in fiat space and time with the Lagrangian ..... dam, 1991.
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ATOM INTERFEROMETRY

MATTER- WAVE INTERFER OMETER S" A SYNTHETIC APPROACH CHRISTIAN

J. B O R D E

Laboratoire de Physique des Lasers, Universit~ Paris-Nord, Villetaneuse, France; and Laboratoire de Gravitation et Cosmologie Relativistes, Universit~ Pierre et Marie Curie, Paris, France

I. Physics of the Generalized Beam Splitter . . . . . . . . . . . . . . . . . . . . . A. The Equivalent Two-Level System and the Effective Hamiltonian . . . . . . . . B. Time-Independent Treatment of the Traveling-Wave Beam Splitter . . . . . . . . C. Scattering Matrix in the Time-Dependent Approach and Propagators between Field Zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II. Architecture of Interferometers . . . . . . . . . . . . . . . . . . . . . . . . . . A. The Two-Zone Ramsey Interferometer . . . . . . . . . . . . . . . . . . . . . B. Multiple-Zone Interferometers . . . . . . . . . . . . . . . . . . . . . . . . III. Sensitivity to Gravitational and Electromagnetic Fields: A Unified Approach through the Dirac Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Covariant Dirac Equations . . . . . . . . . . . . . . . . . . . . . . . . . . B. Gravitational and Electromagnetic Phase Shifts . . . . . . . . . . . . . . . . . IV. Conclusions and Directions of Future Progress . . . . . . . . . . . . . . . . . . .

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I. Physics of the Generalized Beam Splitter T h e key e l e m e n t of m o s t m a t t e r - w a v e i n t e r f e r o m e t e r s is a diffractive b e a m splitter. Ideally, a diffractive b e a m splitter is a scattering potential for the incident particles, which is spatially periodic with a wave vector kerr: V(r) ~ exp(ikeff r + i q~] + c.c. and h e n c e couples two m o d e s I and II differing only by their momenta, either p and p + hk~ff or p and p - hkef f. This c o r r e s p o n d s to the case of the neutron i n t e r f e r o m e t e r [ 1], but also to an a t o m i n t e r f e r o m e t e r using a mechanical grating [2] or off-resonant standing laser waves [ 3 - 5 ] . It is possible to generalize this b e a m splitter potential to time varying potentials such as V(r,t) ~ e x p [ - i ( % f f t - k~ff. r + q~)] + c.c. if the incident particles simultaneously m a y u n d e r g o a c h a n g e of their internal state a----> b, states w h o s e respective energies are E a and E b [ 6 - 9 ] (Fig. 1). If the r e s o n a n t c o n d i t i o n hoJeff = hOgba = E b - E a is satisfied (we e x a m i n e later w h a t h a p p e n s out of resonance), the c h a n g e of internal state p r o v i d e s only an additional label to the m o m e n t u m label for channels I and II: I - (a,0), I I - (b,hkeff). We m a y c o n s i d e r

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Copyright 9 1997 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-092460-9

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C. J. Bord~

I

~,

--> P,Ea

Ea

Eb-Ea+h~ FIG. 1. Generalized beam splitter and Bragg condition. In the ideal case where the splitter is thick enough to have a narrow momentum distribution width Ak