80 research outputs found
Generalized Gaussian wave packet dynamics: Integrable and Chaotic Systems
The ultimate semiclassical wave packet propagation technique is a complex,
time-dependent WBK method known as generalized Gaussian wave packet dynamics
(GGWPD). It requires overcoming many technical difficulties in order to be
carried out fully in practice. In its place roughly twenty years ago,
linearized wave packet dynamics was generalized to methods that include sets of
off-center, real trajectories for both classically integrable and chaotic
dynamical systems that completely capture the dynamical transport. The
connections between those methods and GGWPD are developed in a way that enables
a far more practical implementation of GGWPD. The generally complex saddle
point trajectories at its foundation are found using a multi-dimensional,
Newton-Raphson root search method that begins with the set of off-center, real
trajectories. This is possible because there is a one-to-one correspondence.
The neighboring trajectories associated with each off-center, real trajectory
form a path that crosses a unique saddle; there are exceptions which are
straightforward to identify. The method is applied to the kicked rotor to
demonstrate the accuracy improvement as a function of that comes with
using the saddle point trajectories.Comment: 18 pages, 9 figures, corrected a typo in Eqs. 29,3
A Reduced Dimensional Monte Carlo Method: Preliminary Integrations
A technique for reducing the number of integrals in a Monte Carlo calculation
is introduced. For integrations relying on classical or mean-field trajectories
with local weighting functions, it is possible to integrate analytically at
least half of the integration variables prior to setting up the particular
Monte Carlo calculation of interest, in some cases more. Proper accounting of
invariant phase space structures shows the system's dynamics is reducible into
composite stable and unstable degrees of freedom. Stable degrees of freedom
behave locally in the reduced dimensional phase space exactly as an analogous
integrable system would. Classification of the unstable degrees of freedom is
dependent upon the degree of chaos present in the dynamics. The techniques for
deriving the requisite canonical coordinate transformations are developed and
shown to block diagonalize the stability matrix into irreducible parts. In
doing so, it is demonstrated how to reduce the amount of sampling directions
necessary in a Monte Carlo simulation. The technique is illustrated by
calculating return probabilities and expectation values for different dynamical
regimes of a two-degree-of-freedom coupled quartic oscillator within a
classical Wigner method framework
Semiclassical propagation of coherent states and wave packets: hidden saddles
Semiclassical methods are extremely important in the subjects of wave packet
and coherent state dynamics. Unfortunately, these essentially saddle point
approximations are considered nearly impossible to carry out in detail for
systems with multiple degrees of freedom due to the difficulties of solving the
resulting two-point boundary value problems. However, recent developments have
extended the applicability to a broader range of systems and circumstances. The
most important advances are first to generate a set of real reference
trajectories using appropriately reduced dimensional spaces of initial
conditions, and second to feed that set into a Newton-Raphson search scheme to
locate the complex saddle trajectories. The arguments for this
approach were based mostly on intuition and numerical verification. In this
paper, the methods are put on a firmer theoretical foundation and then extended
to incorporate saddles from Newton-Raphson searches initiated with
real trajectories. This hidden class of saddles is relevant to tunneling-type
processes, but a hidden saddle can sometimes contribute just as much as or more
than an exposed one. The distinctions between hidden and exposed saddles
clarifies the interpretation of what constitutes tunneling for wave packets and
coherent states in the time domain
Partial local density of states from scanning gate microscopy
Scanning gate microscopy images from measurements made in the vicinity of
quantum point contacts were originally interpreted in terms of current flow.
Some recent work has analytically connected the local density of states to
conductance changes in cases of perfect transmission, and at least
qualitatively for a broader range of circumstances. In the present paper, we
show analytically that in any time-reversal invariant system there are
important deviations that are highly sensitive to imperfect transmission.
Nevertheless, the unperturbed partial local density of states can be extracted
from a weakly invasive scanning gate microscopy experiment, provided the
quantum point contact is tuned anywhere on a conductance plateau. A
perturbative treatment in the reflection coefficient shows just how sensitive
this correspondence is to the departure from the quantized conductance value
and reveals the necessity of local averaging over the tip position. It is also
shown that the quality of the extracted partial local density of states
decreases with increasing tip radius.Comment: 16 pages, 9 figure
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