Adventures in Invariant Theory

We provide an introduction to enumerating and constructing invariants of group representations via character methods. The problem is contextualised via two case studies arising from our recent work: entanglement measures, for characterising the structure of state spaces for composite quantum systems; and Markov invariants, a robust alternative to parameter-estimation intensive methods of statistical inference in molecular phylogenetics.Comment: 12 pp, includes supplementary discussion of example

Spatial synchronization and extinction of species under external forcing

We study the interplay between synchronization and extinction of a species. Using a general model we show that under a common external forcing, the species with a quadratic saturation term in the population dynamics first undergoes spatial synchronization and then extinction, thereby avoiding the rescue effect. This is because the saturation term reduces the synchronization time scale but not the extinction time scale. The effect can be observed even when the external forcing acts only on some locations provided there is a synchronizing term in the dynamics. Absence of the quadratic saturation term can help the species to avoid extinction.Comment: 4 pages, 2 figure

Circuit QED scheme for realization of the Lipkin-Meshkov-Glick model

We propose a scheme in which the Lipkin-Meshkov-Glick model is realized within a circuit QED system. An array of N superconducting qubits interacts with a driven cavity mode. In the dispersive regime, the cavity mode is adiabatically eliminated generating an effective model for the qubits alone. The characteristic long-range order of the Lipkin-Meshkov-Glick model is here mediated by the cavity field. For a closed qubit system, the inherent second order phase transition of the qubits is reflected in the intensity of the output cavity field. In the broken symmetry phase, the many-body ground state is highly entangled. Relaxation of the qubits is analyzed within a mean-field treatment. The second order phase transition is lost, while new bistable regimes occur.Comment: 5 pages, 2 figure

Measurement crosstalk between two phase qubits coupled by a coplanar waveguide

We analyze the measurement crosstalk between two flux-biased phase qubits coupled by a resonant coplanar waveguide cavity. After the first qubit is measured, the superconducting phase can undergo damped oscillations resulting in an a.c. voltage that produces a frequency chirped noise signal whose frequency crosses that of the cavity. We show experimentally that the coplanar waveguide cavity acts as a bandpass filter that can significantly reduce the crosstalk signal seen by the second qubit when its frequency is far from the cavity's resonant frequency. We present a simple classical description of the qubit behavior that agrees well with the experimental data. These results suggest that measurement crosstalk between superconducting phase qubits can be reduced by use of linear or possibly nonlinear resonant cavities as coupling elements.Comment: 4 pages, 3 figure

Parametric coupling between macroscopic quantum resonators

Time-dependent linear coupling between macroscopic quantum resonator modes generates both a parametric amplification also known as a {}"squeezing operation" and a beam splitter operation, analogous to quantum optical systems. These operations, when applied properly, can robustly generate entanglement and squeezing for the quantum resonator modes. Here, we present such coupling schemes between a nanomechanical resonator and a superconducting electrical resonator using applied microwave voltages as well as between two superconducting lumped-element electrical resonators using a r.f. SQUID-mediated tunable coupler. By calculating the logarithmic negativity of the partially transposed density matrix, we quantitatively study the entanglement generated at finite temperatures. We also show that characterization of the nanomechanical resonator state after the quantum operations can be achieved by detecting the electrical resonator only. Thus, one of the electrical resonator modes can act as a probe to measure the entanglement of the coupled systems and the degree of squeezing for the other resonator mode.Comment: 15 pages, 4 figures, submitte

Interrelations Between the Neutron's Magnetic Interactions and the Magnetic Aharonov-Bohm Effect

It is proved that the phase shift of a polarized neutron interacting with a spatially uniform time-dependent magnetic field, demonstrates the same physical principles as the magnetic Aharonov-Bohm effect. The crucial role of inert objects is explained, thereby proving the quantum mechanical nature of the effect. It is also proved that the nonsimply connectedness of the field-free region is not a profound property of the system and that it cannot be regarded as a sufficient condition for a nonzero phase shift.Comment: 18 pages, 1 postscript figure, Late

Parallel and divergent morphological adaptations underlying the evolution of jumping ability in ants

Jumping is a rapid locomotory mode widespread in terrestrial organisms. However, it is a rare specialization in ants. Forward jumping has been reported within four distantly related ant genera: Gigantiops, Harpegnathos, Myrmecia, and Odontomachus. The temporal engagement of legs/body parts during jump, however, varies across these genera. It is unknown what morphological adaptations underlie such behaviors and whether jumping in ants is solely driven directly by muscle contraction or additionally relies on elastic recoil mechanism. We investigated the morphological adaptations for jumping behavior by comparing differences in the locomotory musculature between jumping and non-jumping relatives using X-ray micro-CT and 3D morphometrics. We found that the size-specific volumes of the trochanter depressor muscle (scm6) of the middle and hind legs are 3-5 times larger in jumping ants, and that one coxal remotor muscle (scm2) is reduced in volume in the middle and/or hind legs. Notably, the enlargement in the volume of other muscle groups is directly linked to the legs or body parts engaged during the jump. Furthermore, a direct comparison of the muscle architecture revealed two significant differences between jumping vs. non-jumping ants: First, the relative Physiological Cross-Sectional Area (PCSA) of the trochanter depressor muscles of all three legs were larger in jumping ants, except in the front legs of Odontomachus rixosus and Myrmecia nigrocincta; second, the relative muscle fiber length was shorter in jumping ants compared to non-jumping counterparts, except in the front legs of O. rixosus and M. nigrocincta. These results suggest that the difference in relative muscle volume in jumping ants is largely invested in the area (PCSA), and not in fiber length. There was no clear difference in the pennation angle between jumping and non-jumping ants. Additionally, we report that the hind leg length relative to body length was longer in jumping ants. Based on direct comparison of the observed vs. possible work and power output during jumps, we surmise that direct muscle contractions suffice to explain jumping performance in three species, except for O. rixosus, where the lack of data on jumping performance prevents us from drawing definitive conclusions for this particular species. We suggest that increased investment in jumping-relevant musculature is a primary morphological adaptation that separates jumping from non-jumping ants. These results elucidate the common and idiosyncratic morphological changes underlying this rare adaptation in ants. まとぅみ (Okinawan language-Uchinaaguchi) (Japanese) РЕЗЮМЕ (Kazakh) ZUSAMMENFASSUNG (German)