Nonlinear optical probe of tunable surface electrons on a topological insulator

We use ultrafast laser pulses to experimentally demonstrate that the second-order optical response of bulk single crystals of the topological insulator Bi$_2$Se$_3$ is sensitive to its surface electrons. By performing surface doping dependence measurements as a function of photon polarization and sample orientation we show that second harmonic generation can simultaneously probe both the surface crystalline structure and the surface charge of Bi$_2$Se$_3$. Furthermore, we find that second harmonic generation using circularly polarized photons reveals the time-reversal symmetry properties of the system and is surprisingly robust against surface charging, which makes it a promising tool for spectroscopic studies of topological surfaces and buried interfaces

Teaching forest stand dynamics or what happens when you thin your Marigold plantation

Teaching forestry students about forest stand dynamics can be an abstract activity. Very quickly concepts are reduced to mathematical formulae, graphs and diagrams, all with relatively complicated explanations. Alternatively, computer simulation and individual tree models can be used to demonstrate important concepts such as the \u273/2 Power law\u27 of self thinning. Students can also be taken to visit plantations to talk about practical issues of density management and perhaps produce a thinning prescription. However, no single teaching strategy enables students to have \u27hands on\u27 practice at manipulating a real plant population while being able to wait and see the results of their work

Algorithmic Analysis of Qualitative and Quantitative Termination Problems for Affine Probabilistic Programs

In this paper, we consider termination of probabilistic programs with real-valued variables. The questions concerned are: 1. qualitative ones that ask (i) whether the program terminates with probability 1 (almost-sure termination) and (ii) whether the expected termination time is finite (finite termination); 2. quantitative ones that ask (i) to approximate the expected termination time (expectation problem) and (ii) to compute a bound B such that the probability to terminate after B steps decreases exponentially (concentration problem). To solve these questions, we utilize the notion of ranking supermartingales which is a powerful approach for proving termination of probabilistic programs. In detail, we focus on algorithmic synthesis of linear ranking-supermartingales over affine probabilistic programs (APP's) with both angelic and demonic non-determinism. An important subclass of APP's is LRAPP which is defined as the class of all APP's over which a linear ranking-supermartingale exists. Our main contributions are as follows. Firstly, we show that the membership problem of LRAPP (i) can be decided in polynomial time for APP's with at most demonic non-determinism, and (ii) is NP-hard and in PSPACE for APP's with angelic non-determinism; moreover, the NP-hardness result holds already for APP's without probability and demonic non-determinism. Secondly, we show that the concentration problem over LRAPP can be solved in the same complexity as for the membership problem of LRAPP. Finally, we show that the expectation problem over LRAPP can be solved in 2EXPTIME and is PSPACE-hard even for APP's without probability and non-determinism (i.e., deterministic programs). Our experimental results demonstrate the effectiveness of our approach to answer the qualitative and quantitative questions over APP's with at most demonic non-determinism.Comment: 24 pages, full version to the conference paper on POPL 201

Wrinkling of a bilayer membrane

The buckling of elastic bodies is a common phenomenon in the mechanics of solids. Wrinkling of membranes can often be interpreted as buckling under constraints that prohibit large amplitude deformation. We present a combination of analytic calculations, experiments, and simulations to understand wrinkling patterns generated in a bilayer membrane. The model membrane is composed of a flexible spherical shell that is under tension and that is circumscribed by a stiff, essentially incompressible strip with bending modulus B. When the tension is reduced sufficiently to a value \sigma, the strip forms wrinkles with a uniform wavelength found theoretically and experimentally to be \lambda = 2\pi(B/\sigma)^{1/3}. Defects in this pattern appear for rapid changes in tension. Comparison between experiment and simulation further shows that, with larger reduction of tension, a second generation of wrinkles with longer wavelength appears only when B is sufficiently small.Comment: 9 pages, 5 color figure

Theoretical and experimental study of second harmonic generation from the surface of the topological insulator Bi_2Se_3

We develop a theoretical model that describes the second harmonic generation of light from the surface of the topological insulator Bi_2Se_3 and experimentally demonstrate that the technique is sensitive to the surface electrons. By performing a crystal symmetry analysis of Bi_2Se_3 (111) we determine the nonlinear electric susceptibility tensor elements that give rise to second harmonic generation. Using these results, we present a phenomenological model that shows that the relative magnitudes of these tensor elements can be determined by measuring the polarization and intensity of the radiated second harmonic light as a function of the in-plane crystal orientation and incident laser polarization. We describe optical techniques capable of isolating second harmonic light and, using these techniques, we measure the first-order linear optical and second-order nonlinear optical responses as a function of crystal orientation and laser polarization on bulk single crystals of Bi_2Se_3 (111). The experimental results are consistent with our theoretical description. By comparing the data to our theoretical model we determine that a portion of the measured second harmonic light originates from the accumulation region of Bi_2Se_3 (111), which we confirm by performing surface doping-dependent studies. Our results show that second harmonic generation is a promising tool for spectroscopic studies of topological surfaces and buried interfaces

Selective probing of photo-induced charge and spin dynamics in the bulk and surface of a topological insulator

Topological insulators possess completely different spin-orbit coupled bulk and surface electronic spectra that are each predicted to exhibit exotic responses to light. Here we report time-resolved fundamental and second harmonic optical pump-probe measurements on the topological insulator Bi2Se3 to independently measure its photo-induced charge and spin dynamics with bulk and surface selectivity. Our results show that a transient net spin density can be optically induced in both the bulk and surface, which may drive spin transport in topological insulators. By utilizing a novel rotational anisotropy analysis we are able to separately resolve the spin de-polarization, intraband cooling and interband recombination processes following photo-excitation, which reveal that spin and charge degrees of freedom relax on very different time scales owing to strong spin-orbit coupling.Comment: Accepted to Phys. Rev. Let

Completely positive maps with memory

The prevailing description for dissipative quantum dynamics is given by the Lindblad form of a Markovian master equation, used under the assumption that memory effects are negligible. However, in certain physical situations, the master equation is essentially of a non-Markovian nature. This paper examines master equations that possess a memory kernel, leading to a replacement of white noise by colored noise. The conditions under which this leads to a completely positive, trace-preserving map are discussed for an exponential memory kernel. A physical model that possesses such an exponential memory kernel is presented. This model contains a classical, fluctuating environment based on random telegraph signal stochastic variables.Comment: 4 page

Pressure-Coring of the Gas-bearing Devonian Black Shales

A Christensen model pressure-retaining core barrel is being used in the drilling of two EGSP research wells. This effort supported by two other techniques is designed to evaluate the most effective technique to provide meaningful resource assessments. One well has been successfully drilled and the accumulated samples are currently being evaluated. The second well is currently being drilled

Stochastic Invariants for Probabilistic Termination

Termination is one of the basic liveness properties, and we study the termination problem for probabilistic programs with real-valued variables. Previous works focused on the qualitative problem that asks whether an input program terminates with probability~1 (almost-sure termination). A powerful approach for this qualitative problem is the notion of ranking supermartingales with respect to a given set of invariants. The quantitative problem (probabilistic termination) asks for bounds on the termination probability. A fundamental and conceptual drawback of the existing approaches to address probabilistic termination is that even though the supermartingales consider the probabilistic behavior of the programs, the invariants are obtained completely ignoring the probabilistic aspect. In this work we address the probabilistic termination problem for linear-arithmetic probabilistic programs with nondeterminism. We define the notion of {\em stochastic invariants}, which are constraints along with a probability bound that the constraints hold. We introduce a concept of {\em repulsing supermartingales}. First, we show that repulsing supermartingales can be used to obtain bounds on the probability of the stochastic invariants. Second, we show the effectiveness of repulsing supermartingales in the following three ways: (1)~With a combination of ranking and repulsing supermartingales we can compute lower bounds on the probability of termination; (2)~repulsing supermartingales provide witnesses for refutation of almost-sure termination; and (3)~with a combination of ranking and repulsing supermartingales we can establish persistence properties of probabilistic programs. We also present results on related computational problems and an experimental evaluation of our approach on academic examples.Comment: Full version of a paper published at POPL 2017. 20 page