408 research outputs found
The Maslov index and nondegenerate singularities of integrable systems
We consider integrable Hamiltonian systems in R^{2n} with integrals of motion
F = (F_1,...,F_n) in involution. Nondegenerate singularities are critical
points of F where rank dF = n-1 and which have definite linear stability. The
set of nondegenerate singularities is a codimension-two symplectic submanifold
invariant under the flow. We show that the Maslov index of a closed curve is a
sum of contributions +/- 2 from the nondegenerate singularities it is encloses,
the sign depending on the local orientation and stability at the singularities.
For one-freedom systems this corresponds to the well-known formula for the
Poincar\'e index of a closed curve as the oriented difference between the
number of elliptic and hyperbolic fixed points enclosed. We also obtain a
formula for the Liapunov exponent of invariant (n-1)-dimensional tori in the
nondegenerate singular set. Examples include rotationally symmetric n-freedom
Hamiltonians, while an application to the periodic Toda chain is described in a
companion paper.Comment: 27 pages, 1 figure; published versio
Singularities, Lax degeneracies and Maslov indices of the periodic Toda chain
The n-particle periodic Toda chain is a well known example of an integrable
but nonseparable Hamiltonian system in R^{2n}. We show that Sigma_k, the k-fold
singularities of the Toda chain, ie points where there exist k independent
linear relations amongst the gradients of the integrals of motion, coincide
with points where there are k (doubly) degenerate eigenvalues of
representatives L and Lbar of the two inequivalent classes of Lax matrices
(corresponding to degenerate periodic or antiperiodic solutions of the
associated second-order difference equation). The singularities are shown to be
nondegenerate, so that Sigma_k is a codimension-2k symplectic submanifold.
Sigma_k is shown to be of elliptic type, and the frequencies of transverse
oscillations under Hamiltonians which fix Sigma_k are computed in terms of
spectral data of the Lax matrices. If mu(C) is the (even) Maslov index of a
closed curve C in the regular component of R^{2n}, then (-1)^{\mu(C)/2} is
given by the product of the holonomies (equal to +/- 1) of the even- (or odd-)
indexed eigenvector bundles of L and Lmat.Comment: 25 pages; published versio
Magnetic Field Dependence of the Level Spacing of a Small Electron Droplet
The temperature dependence of conductance resonances is used to measure the
evolution with the magnetic field of the average level spacing
of a droplet containing electrons created by lateral confinement of a
two-dimensional electron gas in GaAs. becomes very small (eV) near two critical magnetic fields at which the symmetry of the
droplet changes and these decreases of are predicted by
Hartree-Fock (HF) for charge excitations. Between the two critical fields,
however, the largest measured eV is an order of
magnitude smaller than predicted by HF but comparable to the Zeeman splitting
at this field, which suggests that the spin degrees of freedom are important.
PACS: 73.20.Dx, 73.20.MfComment: 11 pages of text in RevTeX, 4 figures in Postscript (files in the
form of uuencoded compressed tar file
Superposition of photon- and phonon- assisted tunneling in coupled quantum dots
We report on electron transport through an artificial molecule formed by two
tunnel coupled quantum dots, which are laterally confined in a two-dimensional
electron system of an AlGaAs/GaAs heterostructure. Coherent
molecular states in the coupled dots are probed by photon-assisted tunneling
(PAT). Above 10 GHz, we observe clear PAT as a result of the resonance between
the microwave photons and the molecular states. Below 8 GHz, a pronounced
superposition of phonon- and photon-assisted tunneling is observed. Coherent
superposition of molecular states persists under excitation of acoustic
phonons.Comment: 5 pages, 4 figure
Maslov Indices and Monodromy
We prove that for a Hamiltonian system on a cotangent bundle that is
Liouville-integrable and has monodromy the vector of Maslov indices is an
eigenvector of the monodromy matrix with eigenvalue 1. As a corollary the
resulting restrictions on the monodromy matrix are derived.Comment: 6 page
Signatures of Chaos in the Statistical Distribution of Conductance Peaks in Quantum Dots
Analytical expressions for the width and conductance peak distributions of
irregularly shaped quantum dots in the Coulomb blockade regime are presented in
the limits of conserved and broken time-reversal symmetry. The results are
obtained using random matrix theory and are valid in general for any number of
non-equivalent and correlated channels, assuming that the underlying classical
dynamic of the electrons in the dot is chaotic or that the dot is weakly
disordered. The results are expressed in terms of the channel correlation
matrix which for chaotic systems is given in closed form for both point-like
contacts and extended leads. We study the dependence of the distributions on
the number of channels and their correlations. The theoretical distributions
are in good agreement with those computed in a dynamical model of a chaotic
billiard.Comment: 19 pages, RevTex, 11 Postscript figure
Leading off-diagonal contribution to the spectral form factor of chaotic quantum systems
We semiclassically derive the leading off-diagonal correction to the spectral
form factor of quantum systems with a chaotic classical counterpart. To this
end we present a phase space generalization of a recent approach for uniformly
hyperbolic systems (M. Sieber and K. Richter, Phys. Scr. T90, 128 (2001); M.
Sieber, J. Phys. A: Math. Gen. 35, L613 (2002)). Our results coincide with
corresponding random matrix predictions. Furthermore, we study the transition
from the Gaussian orthogonal to the Gaussian unitary ensemble.Comment: 8 pages, 2 figures; J. Phys. A: Math. Gen. (accepted for publication
Non-invasive detection of the evolution of the charge states of a double dot system
Coupled quantum dots are potential candidates for qubit systems in quantum
computing. We use a non-invasive voltage probe to study the evolution of a
coupled dot system from a situation where the dots are coupled to the leads to
a situation where they are isolated from the leads. Our measurements allow us
to identify the movement of electrons between the dots and we can also identify
the presence of a charge trap in our system by detecting the movement of
electrons between the dots and the charge trap. The data also reveals evidence
of electrons moving between the dots via excited states of either the single
dots or the double dot molecule.Comment: Accepted for publication in Phys. Rev. B. 4 pages, 4 figure
Scaling Of The Coulomb Energy Due To Quantum Fluctuations In The Charge Of A Quantum Dot
The charging energy of a quantum dot is measured through the effect of its
potential on the conductance of a second dot. This technique allows a
measurement of the scaling of the dot's charging energy with the conductance of
the tunnel barriers leading to the dot. We find that the charging energy scales
quadratically with the reflection probability of the barriers. In a second
experiment we study the transition from a single to a double-dot which exhibits
a scaling behavior linear in the reflection probability. The observed
power-laws agree with a recent theory.Comment: 5 pages, uuencoded and compressed postscript file, with figure
The Anderson Model out of equilibrium: Time dependent perturbations
The influence of high-frequency fields on quantum transport through a quantum
dot is studied in the low-temperature regime. We generalize the non crossing
approximation for the infinite-U Anderson model to the time-dependent case. The
dc spectral density shows asymmetric Kondo side peaks due to photon-assisted
resonant tunneling. As a consequence we predict an electron-photon pump at zero
bias which is purely based on the Kondo effect. In contrast to the resonant
level model and the time-independent case we observe asymmetric peak amplitudes
in the Coulomb oscillations and the differential conductance versus bias
voltage shows resonant side peaks with a width much smaller than the tunneling
rate. All the effects might be used to clarify the question whether quantum
dots indeed show the Kondo effect.Comment: 13 pages, REVTEX 3.0, 5 figure
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