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 $\Delta\epsilon$ of a droplet containing $\sim 30$ electrons created by lateral confinement of a two-dimensional electron gas in GaAs. $\Delta\epsilon$ becomes very small ($< 30\mu$eV) near two critical magnetic fields at which the symmetry of the droplet changes and these decreases of $\Delta\epsilon$ are predicted by Hartree-Fock (HF) for charge excitations. Between the two critical fields, however, the largest measured $\Delta\epsilon= 100\mu$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 Al$_x$Ga$_{1-x}$As/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

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

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