Weighted Supermembrane Toy Model

A weighted Hilbert space approach to the study of zero-energy states of supersymmetric matrix models is introduced. Applied to a related but technically simpler model, it is shown that the spectrum of the corresponding weighted Hamiltonian simplifies to become purely discrete for sufficient weights. This follows from a bound for the number of negative eigenvalues of an associated matrix-valued Schr\"odinger operator.Comment: 18 pages, 2 figures; to appear in Lett. Math. Phys

Green functions and nonlinear systems: Short time expansion

We show that Green function methods can be straightforwardly applied to nonlinear equations appearing as the leading order of a short time expansion. Higher order corrections can be then computed giving a satisfactory agreement with numerical results. The relevance of these results relies on the possibility of fully exploiting a gradient expansion in both classical and quantum field theory granting the existence of a strong coupling expansion. Having a Green function in this regime in quantum field theory amounts to obtain the corresponding spectrum of the theory.Comment: 7 pages, 3 figures. Version accepted for publication in International Journal of Modern Physics

Mean eigenvalues for simple, simply connected, compact Lie groups

We determine for each of the simple, simply connected, compact and complex Lie groups SU(n), Spin$(4n+2)$ and $E_6$ that particular region inside the unit disk in the complex plane which is filled by their mean eigenvalues. We give analytical parameterizations for the boundary curves of these so-called trace figures. The area enclosed by a trace figure turns out to be a rational multiple of $\pi$ in each case. We calculate also the length of the boundary curve and determine the radius of the largest circle that is contained in a trace figure. The discrete center of the corresponding compact complex Lie group shows up prominently in the form of cusp points of the trace figure placed symmetrically on the unit circle. For the exceptional Lie groups $G_2$, $F_4$ and $E_8$ with trivial center we determine the (negative) lower bound on their mean eigenvalues lying within the real interval $[-1,1]$. We find the rational boundary values -2/7, -3/13 and -1/31 for $G_2$, $F_4$ and $E_8$, respectively.Comment: 12 pages, 8 figure

Witten index, axial anomaly, and Krein's spectral shift function in supersymmetric quantum mechanics

A new method is presented to study supersymmetric quantum mechanics. Using relative scattering techniques, basic relations are derived between Krein’s spectral shift function, the Witten index, and the anomaly. The topological invariance of the spectral shift function is discussed. The power of this method is illustrated by treating various models and calculating explicitly the spectral shift function, the Witten index, and the anomaly. In particular, a complete treatment of the two‐dimensional magnetic field problem is given, without assuming that the magnetic flux is quantized

On the existence of impurity bound excitons in one-dimensional systems with zero range interactions

We consider a three-body one-dimensional Schr\"odinger operator with zero range potentials, which models a positive impurity with charge $\kappa > 0$ interacting with an exciton. We study the existence of discrete eigenvalues as $\kappa$ is varied. On one hand, we show that for sufficiently small $\kappa$ there exists a unique bound state whose binding energy behaves like $\kappa^4$, and we explicitly compute its leading coefficient. On the other hand, if $\kappa$ is larger than some critical value then the system has no bound states

Time-Reversal Symmetry and Universal Conductance Fluctuations in a Driven Two-Level System

In the presence of time-reversal symmetry, quantum interference gives strong corrections to the electric conductivity of disordered systems. The self-interference of an electron wavefunction traveling time-reversed paths leads to effects such as weak localization and universal conductance fluctuations. Here, we investigate the effects of broken time-reversal symmetry in a driven artificial two-level system. Using a superconducting flux qubit, we implement scattering events as multiple Landau-Zener transitions by driving the qubit periodically back and forth through an avoided crossing. Interference between different qubit trajectories give rise to a speckle pattern in the qubit transition rate, similar to the interference patterns created when coherent light is scattered off a disordered potential. Since the scattering events are imposed by the driving protocol, we can control the time-reversal symmetry of the system by making the drive waveform symmetric or asymmetric in time. We find that the fluctuations of the transition rate exhibit a sharp peak when the drive is time-symmetric, similar to universal conductance fluctuations in electronic transport through mesoscopic systems

Evidence for a continuum limit in causal set dynamics

We find evidence for a continuum limit of a particular causal set dynamics which depends on only a single ``coupling constant'' $p$ and is easy to simulate on a computer. The model in question is a stochastic process that can also be interpreted as 1-dimensional directed percolation, or in terms of random graphs.Comment: 24 pages, 19 figures, LaTeX, adjusted terminolog

Dispersion and fidelity in quantum interferometry

We consider Mach-Zehnder and Hong-Ou-Mandel interferometers with nonclassical states of light as input, and study the effect that dispersion inside the interferometer has on the sensitivity of phase measurements. We study in detail a number of different one- and two-photon input states, including Fock, dual Fock, N00N states, and photon pairs from parametric downconversion. Assuming there is a phase shift $\phi_0$ in one arm of the interferometer, we compute the probabilities of measurement outcomes as a function of $\phi_0$, and then compute the Shannon mutual information between $\phi_0$ and the measurements. This provides a means of quantitatively comparing the utility of various input states for determining the phase in the presence of dispersion. In addition, we consider a simplified model of parametric downconversion for which probabilities can be explicitly computed analytically, and which serves as a limiting case of the more realistic downconversion model.Comment: 12 pages, 14 figures. Submitted to Physical Review