256 Atom interferometry
Fig. 11.7An interferometer formed by Raman transitions. As atoms traverse the three interaction regions they experience a
π/2–π–π/2 sequence of Raman pulses that split, deflect and recombine the atomic wavepackets. Each interaction region has two
counter-propagating beams at frequenciesωL1andωL2, as in Fig. 11.6. In this Mach–Zehnder interferometer the eigenstates
with transverse momentumpandp+2kare associated with the different internal atomic states| 1 〉and| 2 〉, respectively;
as indicated at the bottom of the figure,| 1 〉is associated withp 0 (horizontal line) and| 2 〉withp
2 k(slanted line).
Therefore at the output it is only necessary to measure the internal state of the atoms, e.g. by exciting a transition from| 2 〉and
detecting the fluorescence, rather than allowing beams with different momenta to become spatially separated, as in Fig. 11.3.
(The separation of the paths has been exaggerated for clarity.)
the first interaction the atom experiences aπ/2-pulse that puts it into
the superposition|ψ〉={
| 1 ,p〉+eiφ^1 | 2 ,p+2k〉}
. (11.14)
(^16) This phase, and the other similar The phase factor depends on the relative phase of the two laser beams. 16
phases that arise at each interaction,
lead to an offset in the final output
which is not important. But these
phases must remain constant in time,
otherwise the interference ‘washes out’.
These two states separate, as shown in Fig. 11.7, and this first region cor-
responds to a beam splitter for matter waves. After a free-flight through
adistanceLthe atom enters the middle interaction region where it un-
dergoes aπ-pulse that acts on both arms of the interferometer to swap
the states| 1 ,p〉↔| 2 ,p+2k〉. (In this apparatus the transit time for
the atom to pass through the laser beams determines the duration of
the Raman interaction.) The paths come back together after a further
distanceLand the finalπ/2-pulse acts as the beam splitter that mixes
the wavepackets to give interference. The completeπ/2–π–π/2 sequence
gives a Mach–Zehnder interferometer. A comparison of Figs 11.3 and
11.7 shows that the Raman scheme resembles the Mach–Zehnder inter-
ferometer more closely than the three-grating device; the Raman scheme
does not direct any amplitude in unwanted directions and the middle in-
teraction region in the Raman interferometer acts just like a mirror to
change the direction (transverse momentum) of both paths through a
small angle.^17
(^17) In a three-grating interferometer only
a fraction of the amplitude goes in the
required direction. Note, however, that
the simple treatment of standing waves
assumed the ‘thin’ grating approxima-
tion, but often the interaction between
the matter waves and light takes place
over a sufficiently long distance that
Bragg diffraction occurs (as in crys-
tals).
A Raman pulse and standing light wave give the same opening an-
gle between the arms for a given wavelength of laser light and both