1,054 research outputs found
Time Development of Exponentially Small Non-Adiabatic Transitions
Optimal truncations of asymptotic expansions are known to yield
approximations to adiabatic quantum evolutions that are accurate up to
exponentially small errors. In this paper, we rigorously determine the leading
order non--adiabatic corrections to these approximations for a particular
family of two--level analytic Hamiltonian functions. Our results capture the
time development of the exponentially small transition that takes place between
optimal states by means of a particular switching function. Our results confirm
the physics predictions of Sir Michael Berry in the sense that the switching
function for this family of Hamiltonians has the form that he argues is
universal
The Langevin Equation for a Quantum Heat Bath
We compute the quantum Langevin equation (or quantum stochastic differential
equation) representing the action of a quantum heat bath at thermal equilibrium
on a simple quantum system. These equations are obtained by taking the
continuous limit of the Hamiltonian description for repeated quantum
interactions with a sequence of photons at a given density matrix state. In
particular we specialise these equations to the case of thermal equilibrium
states. In the process, new quantum noises are appearing: thermal quantum
noises. We discuss the mathematical properties of these thermal quantum noises.
We compute the Lindblad generator associated with the action of the heat bath
on the small system. We exhibit the typical Lindblad generator that provides
thermalization of a given quantum system.Comment: To appear in J.F.
Determination of Non-Adiabatic Scattering Wave Functions in a Born-Oppenheimer Model
We study non--adiabatic transitions in scattering theory for the time
dependent molecular Schroedinger equation in the Born--Oppenheimer limit. We
assume the electron Hamiltonian has finitely many levels and consider the
propagation of coherent states with high enough total energy. When two of the
electronic levels are isolated from the rest of the electron Hamiltonian's
spectrum and display an avoided crossing, we compute the component of the
nuclear wave function associated with the non--adiabatic transition that is
generated by propagation through the avoided crossing. This component is shown
to be exponentially small in the square of the Born--Oppenheimer parameter, due
to the Landau-Zener mechanism. It propagates asymptotically as a free Gaussian
in the nuclear variables, and its momentum is shifted. The total transition
probability for this transition and the momentum shift are both larger than
what one would expect from a naive approximation and energy conservation
Adiabatic Evolution for Systems with Infinitely many Eigenvalue Crossings
We formulate an adiabatic theorem adapted to models that present an
instantaneous eigenvalue experiencing an infinite number of crossings with the
rest of the spectrum. We give an upper bound on the leading correction terms
with respect to the adiabatic limit. The result requires only differentiability
of the considered spectral projector, and some geometric hypothesis on the
local behaviour of the eigenvalues at the crossings
General Adiabatic Evolution with a Gap Condition
We consider the adiabatic regime of two parameters evolution semigroups
generated by linear operators that are analytic in time and satisfy the
following gap condition for all times: the spectrum of the generator consists
in finitely many isolated eigenvalues of finite algebraic multiplicity, away
from the rest of the spectrum. The restriction of the generator to the spectral
subspace corresponding to the distinguished eigenvalues is not assumed to be
diagonalizable. The presence of eigenilpotents in the spectral decomposition of
the generator forbids the evolution to follow the instantaneous eigenprojectors
of the generator in the adiabatic limit. Making use of superadiabatic
renormalization, we construct a different set of time-dependent projectors,
close to the instantaneous eigeprojectors of the generator in the adiabatic
limit, and an approximation of the evolution semigroup which intertwines
exactly between the values of these projectors at the initial and final times.
Hence, the evolution semigroup follows the constructed set of projectors in the
adiabatic regime, modulo error terms we control
Fundamental solution method applied to time evolution of two energy level systems: exact and adiabatic limit results
A method of fundamental solutions has been used to investigate transitions in
two energy level systems with no level crossing in a real time. Compact
formulas for transition probabilities have been found in their exact form as
well as in their adiabatic limit. No interference effects resulting from many
level complex crossings as announced by Joye, Mileti and Pfister (Phys. Rev.
{\bf A44} 4280 (1991)) have been detected in either case. It is argued that
these results of this work are incorrect. However, some effects of Berry's
phases are confirmed.Comment: LaTeX2e, 23 pages, 8 EPS figures. Style correcte
Semiclassical Dynamics with Exponentially Small Error Estimates
We construct approximate solutions to the time--dependent Schr\"odinger
equation for small values of . If satisfies appropriate analyticity and
growth hypotheses and , these solutions agree with exact solutions up
to errors whose norms are bounded by , for some and
. Under more restrictive hypotheses, we prove that for sufficiently
small implies the norms of the errors are bounded
by , for some , and
A Time-Dependent Born-Oppenheimer Approximation with Exponentially Small Error Estimates
We present the construction of an exponentially accurate time-dependent
Born-Oppenheimer approximation for molecular quantum mechanics. We study
molecular systems whose electron masses are held fixed and whose nuclear masses
are proportional to , where is a small expansion
parameter. By optimal truncation of an asymptotic expansion, we construct
approximate solutions to the time-dependent Schr\"odinger equation that agree
with exact normalized solutions up to errors whose norms are bounded by \ds C
\exp(-\gamma/\epsilon^2), for some C and
Exponentially Accurate Semiclassical Dynamics: Propagation, Localization, Ehrenfest Times, Scattering and More General States
We prove six theorems concerning exponentially accurate semiclassical quantum
mechanics. Two of these theorems are known results, but have new proofs. Under
appropriate hypotheses, they conclude that the exact and approximate dynamics
of an initially localized wave packet agree up to exponentially small errors in
for finite times and for Ehrenfest times. Two other theorems state that
for such times the wave packets are localized near a classical orbit up to
exponentially small errors. The fifth theorem deals with infinite times and
states an exponentially accurate scattering result. The sixth theorem provides
extensions of the other five by allowing more general initial conditions
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