Theory of nonadiabatic transition for general twostate curve crossing problems. I. Nonadiabatic tunneling caseThe squeezed state approach of the semiclassical limit of the timedependent Schrödinger equationWe show rigorously, within the two-state approximation, that in the semicJassicallimit h->O a nonadiabatic transition induced by an analytic time-dependent Hamiltonian is localized to the vicinity of a complex crossing of the two adiabatic potential curves, with transition amplitude independent of the nonadiabatic coupling and given by a simple formula of A. M. Dykhne.

S U M M A R YExamples of Grafenberg-array data showing anomalous P-waves which typically arrive 3-5s after the direct P-wave and which have a slowness 0.7-0.8 sdeg-' smaller than direct P are presented. This additional phase is most frequently observed for events located in the NE portion of the southern Kurile Island subduction zone 73"-80" from Grafenberg, but systematically disappears for events in the SW portion of this zone.Because of the magnitude of the slowness difference, these observations cannot be attributed to a complex source rupture process nor to multipathing through the descending slab. Likewise, they may not be accounted for by near-receiver structure because these phases are not seen for all Kurile events. If present they appear at all stations of the array but they follow direct P too closely to be a multiple from the Moho. Therefore, we conclude they are very likely caused by lower mantle velocity structure.The most likely explanation is the presence of a P velocity jump of about 3 per cent approximately 290 km above the core-mantle boundary, since such a reflector in the lowermost mantle not only gives a good fit of traveltimes and slowness but is also able to model the waveform and the amplitudes of this additional P phase. The distribution of bounce points on this reflector for the Kurile events indicates a lateral extension of this velocity anomaly under northern Siberia of about 150 km by at least 200 km. The best fitting S-wave model has a reflector in the same depth, but the velocity contrast seems to be only about 2 per cent suggesting a different behaviour of the P and S velocity in D".Few events from other regions in this distance range are suitable for a definitive analysis of this kind. From among this group some observations indicate a lower mantle anomaly under the Lomonosow Ridge and under northern Greenland; but since the lower mantle under western Siberia, northern Novaya Zemlya, the Azores Islands region and the USSR-Afghanistan border region does not produce an additional phase in the Grafenberg recordings it is very unlikely that such a velocity anomaly in the lowermost mantle is a global feature.

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The quantum mechanical propagator of a massive particle in a linear gravitational potential derived already in 1927 by Earle H. Kennard [2, 3] contains a phase that scales with the third power of the time T during which the particle experiences the corresponding force. Since in conventional atom interferometers the internal atomic states are all exposed to the same acceleration a, this T 3 -phase cancels out and the interferometer phase scales as T 2 . In contrast, by applying an external magnetic field we prepare two different accelerations a 1 and a 2 for two internal states of the atom, which translate themselves into two different cubic phases and the resulting interferometer phase scales as T 3 . We present the theoretical background for, and summarize our progress towards experimentally realizing such a novel atom interferometer.

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