Spectroscopic studies in open quantum systems

The spectroscopic properties of an open quantum system are determined by the eigenvalues and eigenfunctions of an effective Hamiltonian H consisting of the Hamiltonian H_0 of the corresponding closed system and a non-Hermitian correction term W arising from the interaction via the continuum of decay channels. The eigenvalues E_R of H are complex. They are the poles of the S-matrix and provide both the energies and widths of the states. We illustrate the interplay between Re(H) and Im(H) by means of the different interference phenomena between two neighboured resonance states. Level repulsion along the real axis appears if the interaction is caused mainly by Re(H) while a bifurcation of the widths appears if the interaction occurs mainly due to Im(H). We then calculate the poles of the S-matrix and the corresponding wavefunctions for a rectangular microwave resonator with a scatter as a function of the area of the resonator as well as of the degree of opening to a guide. The calculations are performed by using the method of exterior complex scaling. Re(W) and Im(W) cause changes in the structure of the wavefunctions which are permanent, as a rule. At full opening to the lead, short-lived collective states are formed together with long-lived trapped states. The wavefunctions of the short-lived states at full opening to the lead are very different from those at small opening. The resonance picture obtained from the microwave resonator shows all the characteristic features known from the study of many-body systems in spite of the absence of two-body forces. The poles of the S-matrix determine the conductance of the resonator. Effects arising from the interplay between resonance trapping and level repulsion along the real axis are not involved in the statistical theory.Comment: The six jpg files are not included in the tex-fil

Fast readout of a single Cooper-pair box using its quantum capacitance

We have fabricated a single Cooper-pair box (SCB) together with an on-chip lumped element resonator. By utilizing the quantum capacitance of the SCB, its state can be read out by detecting the phase of a radio-frequency (rf) signal reflected off the resonator. The resonator was optimized for fast readout. By studying quasiparticle tunneling events in the SCB, we have characterized the performance of the readout and found that we can perform a single shot parity measurement in approximately 50 ns. This is an order of magnitude faster than previously reported measurements.Comment: 7 pages, 5 figure

Influence of surface roughness on superhydrophobicity

Superhydrophobic surfaces, with liquid contact angle theta greater than 150 degree, have important practical applications ranging from self-cleaning window glasses, paints, and fabrics to low-friction surfaces. Many biological surfaces, such as the lotus leaf, have hierarchically structured surface roughness which is optimized for superhydrophobicity through natural selection. Here we present a molecular dynamics study of liquid droplets in contact with self-affine fractal surfaces. Our results indicate that the contact angle for nanodroplets depends strongly on the root-mean-square surface roughness amplitude but is nearly independent of the fractal dimension D_f of the surface.Comment: 5 Pages, 6 figures. Minimal changes with respect to the previous versio

Contact mechanics with adhesion: Interfacial separation and contact area

We study the adhesive contact between elastic solids with randomly rough, self affine fractal surfaces. We present molecular dynamics (MD) simulation results for the interfacial stress distribution and the wall-wall separation. We compare the MD results for the relative contact area and the average interfacial separation, with the prediction of the contact mechanics theory of Persson. We find good agreement between theory and the simulation results. We apply the theory to the system studied by Benz et al. involving polymer in contact with polymer, but in this case the adhesion gives only a small modification of the interfacial separation as a function of the squeezing pressure.Comment: 5 pages, 4 figure

Molecular dynamics study of contact mechanics: contact area and interfacial separation from small to full contact

We report a molecular dynamics study of the contact between a rigid solid with a randomly rough surface and an elastic block with a flat surface. We study the contact area and the interfacial separation from small contact (low load) to full contact (high load). For small load the contact area varies linearly with the load and the interfacial separation depends logarithmically on the load. For high load the contact area approaches to the nominal contact area (i.e., complete contact), and the interfacial separation approaches to zero. The present results may be very important for soft solids, e.g., rubber, or for very smooth surfaces, where complete contact can be reached at moderate high loads without plastic deformation of the solids.Comment: 4 pages,5 figure

Eisenstein series and automorphic representations

We provide an introduction to the theory of Eisenstein series and automorphic forms on real simple Lie groups G, emphasising the role of representation theory. It is useful to take a slightly wider view and define all objects over the (rational) adeles A, thereby also paving the way for connections to number theory, representation theory and the Langlands program. Most of the results we present are already scattered throughout the mathematics literature but our exposition collects them together and is driven by examples. Many interesting aspects of these functions are hidden in their Fourier coefficients with respect to unipotent subgroups and a large part of our focus is to explain and derive general theorems on these Fourier expansions. Specifically, we give complete proofs of the Langlands constant term formula for Eisenstein series on adelic groups G(A) as well as the Casselman--Shalika formula for the p-adic spherical Whittaker function associated to unramified automorphic representations of G(Q_p). In addition, we explain how the classical theory of Hecke operators fits into the modern theory of automorphic representations of adelic groups, thereby providing a connection with some key elements in the Langlands program, such as the Langlands dual group LG and automorphic L-functions. Somewhat surprisingly, all these results have natural interpretations as encoding physical effects in string theory. We therefore also introduce some basic concepts of string theory, aimed toward mathematicians, emphasising the role of automorphic forms. In particular, we provide a detailed treatment of supersymmetry constraints on string amplitudes which enforce differential equations of the same type that are satisfied by automorphic forms. Our treatise concludes with a detailed list of interesting open questions and pointers to additional topics which go beyond the scope of this book.Comment: 326 pages. Detailed and example-driven exposition of the subject with highlighted applications to string theory. v2: 375 pages. Substantially extended and small correction