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This dissertation discusses the Landau -Zener (LZ ) theory and its application in
noisy environments and in many -body systems . The first project considers the effect
of fast quantum noise on LZ transitions . There are two important time intervals
separated by the characteristic LZ time . For each interval we derive and solve the
evolution equation , and match the solutions at the boundaries to get a complete
solution . Outside the LZ time interval , we derive the master equation , which differs
from the classical equation by a quantum commutation term . Inside the LZ time
interval , the mixed longitudinal -transverse noise correlation renormalizes the LZ gap
and the system evolves according to the renormalized LZ gap . In the extreme quantum
regime at zero temperature our theory gives a beautiful result which coincides
with that of other authors . Our initial attempts to solve two experimental puzzles
- an isotope effect and the quantized hysteresis curve of a single molecular magnet -
are also discussed .
The second project considers an ultracold dilute Fermi gas in a magnetic field
sweeping across the broad Feshbach resonance . The broad resonance condition allows
us to use the single mode approximation and to neglect the energy dispersion of the
fermions . We then propose the Global Spin Model Hamiltonian , whose ground state
we solve exactly , which yields the static limit properties of the BEC -BCS crossover . We also study the dynamics of the Global Spin Model by converting it to a LZ
problem . The resulting molecular production from the initial fermions is described
by a LZ -like formula with a strongly renormalized LZ gap that is independent of the
initial fermion density . We predict that molecular production during a field -sweep
strongly depends on the initial value of magnetic field . We predict that in the inverse
process of molecular dissociation , immediately after the sweeping stops there appear
Cooper pairs with parallel electronic spins and opposite momenta . |
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