On invariant Schreier structures

Schreier graphs, which possess both a graph structure and a Schreier structure (an edge-labeling by the generators of a group), are objects of fundamental importance in group theory and geometry. We study the Schreier structures with which unlabeled graphs may be endowed, with emphasis on structures which are invariant in some sense (e.g. conjugation-invariant, or sofic). We give proofs of a number of "folklore" results, such as that every regular graph of even degree admits a Schreier structure, and show that, under mild assumptions, the space of invariant Schreier structures over a given invariant graph structure is very large, in that it contains uncountably many ergodic measures. Our work is directly connected to the theory of invariant random subgroups, a field which has recently attracted a great deal of attention.Comment: 16 pages, added references and figure, to appear in L'Enseignement Mathematiqu

The boundary action of a sofic random subgroup of the free group

We prove that the boundary action of a sofic random subgroup of a finitely generated free group is conservative. This addresses a question asked by Grigorchuk, Kaimanovich, and Nagnibeda, who studied the boundary actions of individual subgroups of the free group. Following their work, we also investigate the cogrowth and various limit sets associated to sofic random subgroups. We make heavy use of the correspondence between subgroups and their Schreier graphs, and central to our approach is an investigation of the asymptotic density of a given set inside of large neighborhoods of the root of a sofic random Schreier graph.Comment: 21 pages, 2 figures, made minor corrections, to appear in Groups, Geometry, and Dynamic

Coherent ultrafast torsional motion and isomerization of a biomimetic dipolar photoswitch

Femtosecond fluorescence up-conversion, UV-Vis and IR transient absorption spectroscopy are used to study the photo-isomerization dynamics of a new type of zwitterionic photoswitch based on a N-alkylated indanylidene pyrroline Schiff base framework (ZW-NAIP). The system is biomimetic, as it mimics the photophysics of retinal, in coupling excited state charge translocation and isomerization. While the fluorescence lifetime is 140 fs, excited state absorption persists over 230 fs in the form of a vibrational wavepacket according to twisting of the isomerizing double bond. After a short "dark'' time window in the UV-visible spectra, which we associate with the passage through a conical intersection (CI), the wavepacket appears on the ground state potential energy surface, as evidenced by the transient mid-IR data. This allows for a precise timing of the photoreaction all the way from the initial Franck-Condon region, through the CI and into both ground state isomers, until incoherent vibrational relaxation dominates the dynamics. The photo-reaction dynamics remarkably follow those observed for retinal in rhodopsin, with the additional benefit that in ZW-NAIP the conformational change reverses the zwitterion dipole moment direction. Last, the pronounced low-frequency coherences make these molecules ideal systems for investigating wavepacket dynamics in the vicinity of a CI and for coherent control experiments

Quantum state preparation and macroscopic entanglement in gravitational-wave detectors

Long-baseline laser-interferometer gravitational-wave detectors are operating at a factor of 10 (in amplitude) above the standard quantum limit (SQL) within a broad frequency band. Such a low classical noise budget has already allowed the creation of a controlled 2.7 kg macroscopic oscillator with an effective eigenfrequency of 150 Hz and an occupation number of 200. This result, along with the prospect for further improvements, heralds the new possibility of experimentally probing macroscopic quantum mechanics (MQM) - quantum mechanical behavior of objects in the realm of everyday experience - using gravitational-wave detectors. In this paper, we provide the mathematical foundation for the first step of a MQM experiment: the preparation of a macroscopic test mass into a nearly minimum-Heisenberg-limited Gaussian quantum state, which is possible if the interferometer's classical noise beats the SQL in a broad frequency band. Our formalism, based on Wiener filtering, allows a straightforward conversion from the classical noise budget of a laser interferometer, in terms of noise spectra, into the strategy for quantum state preparation, and the quality of the prepared state. Using this formalism, we consider how Gaussian entanglement can be built among two macroscopic test masses, and the performance of the planned Advanced LIGO interferometers in quantum-state preparation

Searching for a Stochastic Background of Gravitational Waves with LIGO

The Laser Interferometer Gravitational-wave Observatory (LIGO) has performed the fourth science run, S4, with significantly improved interferometer sensitivities with respect to previous runs. Using data acquired during this science run, we place a limit on the amplitude of a stochastic background of gravitational waves. For a frequency independent spectrum, the new limit is $\Omega_{\rm GW} < 6.5 \times 10^{-5}$. This is currently the most sensitive result in the frequency range 51-150 Hz, with a factor of 13 improvement over the previous LIGO result. We discuss complementarity of the new result with other constraints on a stochastic background of gravitational waves, and we investigate implications of the new result for different models of this background.Comment: 37 pages, 16 figure

An artificial molecular switch that mimics the visual pigment and completes its photocycle in picoseconds

Single molecules that act as light-energy transducers (e.g., converting the energy of a photon into atomic-level mechanical motion) are examples of minimal molecular devices. Here, we focus on a molecular switch designed by merging a conformationally locked diarylidene skeleton with a retinal-like Schiff base and capable of mimicking, in solution, different aspects of the transduction of the visual pigment Rhodopsin. Complementary ab initio multiconfigurational quantum chemistry-based computations and time-resolved spectroscopy are used to follow the light-induced isomerization of the switch in methanol. The results show that, similar to rhodopsin, the isomerization occurs on a 0.3-ps time scale and is followed by <10-ps cooling and solvation. The entire (2-photon-powered) switch cycle was traced by following the evolution of its infrared spectrum. These measurements indicate that a full cycle can be completed within 20 ps