302 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
Correlations between Ground and Excited State Spectra of a Quantum Dot
We have studied the ground and excited state spectra of a semiconductor
quantum dot for successive numbers of electron occupancy using linear and
nonlinear magnetoconductance measurements. We present the first observation of
direct correlation between the mth excited state of the N electron system and
the ground state of the N+m electron system for m up to 4. Results are
consistent with a non-spin-degenerate single particle picture of the filling of
levels. Electron-electron interaction effects are also observed as a
perturbation to this model. Magnetoconductance fluctuations of ground states
are shown as anticrossings where wavefunction characteristics are exchanged
between adjacent levels.Comment: 8 pages pdf; gzipped ps available at
http://www-leland.stanford.edu/group/MarcusLab/grouppubs.htm
Single Electron Transistors
Contains description of one research project.Joint Services Electronics Program Contract DAAL03-89-C-0001Joint Services Electronics Program Contract DAAL03-92-C-0001National Science Foundation Grant ECS 88-1325
Observation of Quantum Fluctuations of Charge on a Quantum Dot
We have incorporated an aluminum single electron transistor directly into the
defining gate structure of a semiconductor quantum dot, permitting precise
measurement of the charge in the dot. Voltage biasing a gate draws charge from
a reservoir into the dot through a single point contact. The charge in the dot
increases continuously for large point contact conductance and in a step-like
manner in units of single electrons with the contact nearly closed. We measure
the corresponding capacitance lineshapes for the full range of point contact
conductances. The lineshapes are described well by perturbation theory and not
by theories in which the dot charging energy is altered by the barrier
conductance.Comment: Revtex, 5 pages, 3 figures, few minor corrections to the reference
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
Zero-bias anomalies and boson-assisted tunneling through quantum dots
We study resonant tunneling through a quantum dot with one degenerate level
in the presence of a strong Coulomb repulsion and a bosonic environment. Using
a real-time approach we calculate the spectral density and the nonlinear
current within a conserving approximation. The spectral density shows a
multiplet of Kondo peaks split by the transport voltage and boson frequencies.
As a consequence we find a zero-bias anomaly in the differential conductance
which can show a local maximum or minimum depending on the level position. The
results are compared with recent experiments.Comment: 4 pages, revtex, 5 postscript figures, submitted to Phys. Rev. Let
Single Electron Transistors
Contains description of one research project.Joint Services Electronics Program Contract DAAL03-89-C-0001National Science Foundation Grant ECS 88-1325
Singularities of bi-Hamiltonian systems
We study the relationship between singularities of bi-Hamiltonian systems and
algebraic properties of compatible Poisson brackets. As the main tool, we
introduce the notion of linearization of a Poisson pencil. From the algebraic
viewpoint, a linearized Poisson pencil can be understood as a Lie algebra with
a fixed 2-cocycle. In terms of such linearizations, we give a criterion for
non-degeneracy of singular points of bi-Hamiltonian systems and describe their
types
Measuring Temperature Gradients over Nanometer Length Scales
When a quantum dot is subjected to a thermal gradient, the temperature of
electrons entering the dot can be determined from the dot's thermocurrent if
the conductance spectrum and background temperature are known. We demonstrate
this technique by measuring the temperature difference across a 15 nm quantum
dot embedded in a nanowire. This technique can be used when the dot's energy
states are separated by many kT and will enable future quantitative
investigations of electron-phonon interaction, nonlinear thermoelectric
effects, and the effciency of thermoelectric energy conversion in quantum dots.Comment: 6 pages, 5 figure
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