174 research outputs found
Symplectic evolution of Wigner functions in markovian open systems
The Wigner function is known to evolve classically under the exclusive action
of a quadratic hamiltonian. If the system does interact with the environment
through Lindblad operators that are linear functions of position and momentum,
we show that the general evolution is the convolution of the classically
evolving Wigner function with a phase space gaussian that broadens in time. We
analyze the three generic cases of elliptic, hyperbolic and parabolic
Hamiltonians. The Wigner function always becomes positive in a definite time,
which is shortest in the hyperbolic case. We also derive an exact formula for
the evolving linear entropy as the average of a narrowing gaussian taken over a
probability distribution that depends only on the initial state. This leads to
a long time asymptotic formula for the growth of linear entropy.Comment: this new version treats the dissipative cas
Decoherence of Semiclassical Wigner Functions
The Lindblad equation governs general markovian evolution of the density
operator in an open quantum system. An expression for the rate of change of the
Wigner function as a sum of integrals is one of the forms of the Weyl
representation for this equation. The semiclassical description of the Wigner
function in terms of chords, each with its classically defined amplitude and
phase, is thus inserted in the integrals, which leads to an explicit
differential equation for the Wigner function. All the Lindblad operators are
assumed to be represented by smooth phase space functions corresponding to
classical variables. In the case that these are real, representing hermitian
operators, the semiclassical Lindblad equation can be integrated. There results
a simple extension of the unitary evolution of the semiclassical Wigner
function, which does not affect the phase of each chord contribution, while
dampening its amplitude. This decreases exponentially, as governed by the time
integral of the square difference of the Lindblad functions along the classical
trajectories of both tips of each chord. The decay of the amplitudes is shown
to imply diffusion in energy for initial states that are nearly pure.
Projecting the Wigner function onto an orthogonal position or momentum basis,
the dampening of long chords emerges as the exponential decay of off-diagonal
elements of the density matrix.Comment: 23 pg, 2 fi
Semiclassical Evolution of Dissipative Markovian Systems
A semiclassical approximation for an evolving density operator, driven by a
"closed" hamiltonian operator and "open" markovian Lindblad operators, is
obtained. The theory is based on the chord function, i.e. the Fourier transform
of the Wigner function. It reduces to an exact solution of the Lindblad master
equation if the hamiltonian operator is a quadratic function and the Lindblad
operators are linear functions of positions and momenta.
Initially, the semiclassical formulae for the case of hermitian Lindblad
operators are reinterpreted in terms of a (real) double phase space, generated
by an appropriate classical double Hamiltonian. An extra "open" term is added
to the double Hamiltonian by the non-hermitian part of the Lindblad operators
in the general case of dissipative markovian evolution. The particular case of
generic hamiltonian operators, but linear dissipative Lindblad operators, is
studied in more detail. A Liouville-type equivariance still holds for the
corresponding classical evolution in double phase, but the centre subspace,
which supports the Wigner function, is compressed, along with expansion of its
conjugate subspace, which supports the chord function.
Decoherence narrows the relevant region of double phase space to the
neighborhood of a caustic for both the Wigner function and the chord function.
This difficulty is avoided by a propagator in a mixed representation, so that a
further "small-chord" approximation leads to a simple generalization of the
quadratic theory for evolving Wigner functions.Comment: 33 pages - accepted to J. Phys.
On the classical-quantum correspondence for the scattering dwell time
Using results from the theory of dynamical systems, we derive a general
expression for the classical average scattering dwell time, tau_av. Remarkably,
tau_av depends only on a ratio of phase space volumes. We further show that,
for a wide class of systems, the average classical dwell time is not in
correspondence with the energy average of the quantum Wigner time delay.Comment: 5 pages, 1 figur
Significance of Ghost Orbit Bifurcations in Semiclassical Spectra
Gutzwiller's trace formula for the semiclassical density of states in a
chaotic system diverges near bifurcations of periodic orbits, where it must be
replaced with uniform approximations. It is well known that, when applying
these approximations, complex predecessors of orbits created in the bifurcation
("ghost orbits") can produce pronounced signatures in the semiclassical spectra
in the vicinity of the bifurcation. It is the purpose of this paper to
demonstrate that these ghost orbits themselves can undergo bifurcations,
resulting in complex, nongeneric bifurcation scenarios. We do so by studying an
example taken from the Diamagnetic Kepler Problem, viz. the period quadrupling
of the balloon orbit. By application of normal form theory we construct an
analytic description of the complete bifurcation scenario, which is then used
to calculate the pertinent uniform approximation. The ghost orbit bifurcation
turns out to produce signatures in the semiclassical spectrum in much the same
way as a bifurcation of real orbits would.Comment: 20 pages, 6 figures, LATEX (IOP style), submitted to J. Phys.
Semiclassical theory for small displacements
Characteristic functions contain complete information about all the moments
of a classical distribution and the same holds for the Fourier transform of the
Wigner function: a quantum characteristic function, or the chord function.
However, knowledge of a finite number of moments does not allow for accurate
determination of the chord function. For pure states this provides the overlap
of the state with all its possible rigid translations (or displacements). We
here present a semiclassical approximation of the chord function for large
Bohr-quantized states, which is accurate right up to a caustic, beyond which
the chord function becomes evanescent. It is verified to pick out blind spots,
which are displacements for zero overlaps. These occur even for translations
within a Planck area of the origin. We derive a simple approximation for the
closest blind spots, depending on the Schroedinger covariance matrix, which is
verified for Bohr-quantized states.Comment: 16 pages, 4 figures
Uniform approximations for non-generic bifurcation scenatios including bifurcations of ghost orbits
Gutzwiller's trace formula allows interpreting the density of states of a
classically chaotic quantum system in terms of classical periodic orbits. It
diverges when periodic orbits undergo bifurcations, and must be replaced with a
uniform approximation in the vicinity of the bifurcations. As a characteristic
feature, these approximations require the inclusion of complex ``ghost
orbits''. By studying an example taken from the Diamagnetic Kepler Problem,
viz. the period-quadrupling of the balloon-orbit, we demonstrate that these
ghost orbits themselves can undergo bifurcations, giving rise to non-generic
complicated bifurcation scenarios. We extend classical normal form theory so as
to yield analytic descriptions of both bifurcations of real orbits and ghost
orbit bifurcations. We then show how the normal form serves to obtain a uniform
approximation taking the ghost orbit bifurcation into account. We find that the
ghost bifurcation produces signatures in the semiclassical spectrum in much the
same way as a bifurcation of real orbits does.Comment: 56 pages, 21 figure, LaTeX2e using amsmath, amssymb, epsfig, and
rotating packages. To be published in Annals of Physic
Noise models for superoperators in the chord representation
We study many-qubit generalizations of quantum noise channels that can be
written as an incoherent sum of translations in phase space. Physical
description in terms of the spectral properties of the superoperator and the
action in phase space are provided. A very natural description of decoherence
leading to a preferred basis is achieved with diffusion along a phase space
line. The numerical advantages of using the chord representation are
illustrated in the case of coarse-graining noise.Comment: 8 pages, 5 .ps figures (RevTeX4). Submitted to Phys. Rev. A. minor
changes made, according to referee suggestion
Singularities of equidistants and global centre symmetry sets of Lagrangian submanifolds
We define the Global Centre Symmetry set (GCS) of a smooth closed
m-dimensional submanifold M of R^n, , which is an affinely invariant
generalization of the centre of a k-sphere in R^{k+1}. The GCS includes both
the centre symmetry set defined by Janeczko and the Wigner caustic defined by
Berry. We develop a new method for studying generic singularities of the GCS
which is suited to the case when M is lagrangian in R^{2m} with canonical
symplectic form. The definition of the GCS, which slightly generalizes one by
Giblin and Zakalyukin, is based on the notion of affine equidistants, so, we
first study singularities of affine equidistants of Lagrangian submanifolds,
classifying all the stable ones. Then, we classify the affine-Lagrangian stable
singularities of the GCS of Lagrangian submanifolds and show that, already for
smooth closed convex curves in R^2, many singularities of the GCS which are
affine stable are not affine-Lagrangian stable.Comment: 26 pages, 2 figure
Classical orbit bifurcation and quantum interference in mesoscopic magnetoconductance
We study the magnetoconductance of electrons through a mesoscopic channel
with antidots. Through quantum interference effects, the conductance maxima as
functions of the magnetic field strength and the antidot radius (regulated by
the applied gate voltage) exhibit characteristic dislocations that have been
observed experimentally. Using the semiclassical periodic orbit theory, we
relate these dislocations directly to bifurcations of the leading classes of
periodic orbits.Comment: 4 pages, including 5 figures. Revised version with clarified
discussion and minor editorial change
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