Fungible dynamics: There are only two types of entangling multiple-qubit interactions

What interactions are sufficient to simulate arbitrary quantum dynamics in a composite quantum system? It has been shown that all two-body Hamiltonian evolutions can be simulated using any fixed two-body entangling n-qubit Hamiltonian and fast local unitaries. By entangling we mean that every qubit is coupled to every other qubit, if not directly, then indirectly via intermediate qubits. We extend this study to the case where interactions may involve more than two qubits at a time. We find necessary and sufficient conditions for an arbitrary n-qubit Hamiltonian to be dynamically universal, that is, able to simulate any other Hamiltonian acting on n qubits, possibly in an inefficient manner. We prove that an entangling Hamiltonian is dynamically universal if and only if it contains at least one coupling term involving an even number of interacting qubits. For odd entangling Hamiltonians, i.e., Hamiltonians with couplings that involve only an odd number of qubits, we prove that dynamic universality is possible on an encoded set of n-1 logical qubits. We further prove that an odd entangling Hamiltonian can simulate any other odd Hamiltonian and classify the algebras that such Hamiltonians generate. Thus, our results show that up to local unitary operations, there are only two fundamentally different types of entangling Hamiltonian on n qubits. We also demonstrate that, provided the number of qubits directly coupled by the Hamiltonian is bounded above by a constant, our techniques can be made efficient.

Quantum dynamics of a model for two Josephson-coupled Bose-Einstein condensates

In this work, we investigate the quantum dynamics of a model for two singlemode Bose-Einstein condensates which are coupled via Josephson tunnelling. Using direct numerical diagonalization of the Hamiltonian, we compute the time evolution of the expectation value for the relative particle number across a wide range of couplings. Our analysis shows that the system exhibits rich and complex behaviours varying between harmonic and non-harmonic oscillations, particularly around the threshold coupling between the delocalized and selftrapping phases. We show that these behaviours are dependent on both the initial state of the system and regime of the coupling. In addition, a study of the dynamics for the variance of the relative particle number expectation and the entanglement for different initial states is presented in detail.

Vibrational dynamics and structure of natural feldspars

The present study gives a contribution to the knowledge on the Na-feldspar and plagioclases, extending the database of the Raman spectra of plagioclases with different chemical compositions and structural orders. This information may be used for the future planetary explorations by “rovers”, for the investigation of ceramics nanocrystal materials and for the mineralogical phase identification in sediments. Na-feldspar and plagioclase solid solution have been investigated by Raman spectroscopy in order to determine the relationships between the vibrational changes and the plagioclase crystal chemistry and structure. We focused on the Raman micro-spectroscopy technique, being a non-destructive method, suited for contactless analysis with high spatial resolution. Chemical and structural analyses have been performed on natural samples to test the usefulness of Raman spectroscopy as a tool in the study of the pressure-induced structural deformations, the disordering processes due to change in the Al-Si distribution in the tetrahedral sites and, finally, in the determination of the anorthitic content (Anx) in plagioclase minerals. All the predicted 39 Ag Raman active modes have been identified and assigned to specific patterns of atomic vibrational motion. A detailed comparison between experimental and computed Raman spectra has been performed and previous assignments have been revised, solving some discrepancies reported in recent literature. The ab initio calculation at the hybrid HF/DFT level with the WC1LYP Hamiltonian has proven to give excellent agreement between calculated and experimentally measured Raman wavenumbers and intensities in triclinic minerals. A short digression on the 36 infrared active modes of Na-feldspar has been done too. The identification of all 39 computed Raman modes in the experimentally measured spectra of the fully ordered Na-feldspar, known as low albite, along with the detailed description of each vibrational mode, has been essential to extend the comparative analysis to the high pressure and high temperature structural forms of albite, which reflect the physical–chemical conditions of the hosting rocks. The understanding of feldspar structure response to pressure and temperature is crucial in order to constrain crustal behaviour. The compressional behaviour of the Na-feldspar has been investigated for the first time by Raman spectroscopy. The absence of phase transitions and the occurrence of two secondary compression mechanisms acting at different pressures have been confirmed. Moreover, Raman data suggest that the internal structural changes are confined to a small pressure interval, localized around 6 GPa, not spread out from 4 to 8 GPa as suggested by previous X-rays studies on elasticity. The dominant compression mechanisms act via tetrahedral tilting, while the T-O bond lengths remain nearly constant at moderate compressional regimes. At the spectroscopic level, this leads to the strong pressure dependencies of T-O-T bending modes, as found for the four modes at 478, 508, 578 and 815 cm-1. The Al-Si distribution in the tetrahedral sites affects also the Raman spectrum of Na-feldspar. In particular, peak broadening is more sensitive than peak position to changes in the degree of order. Raman spectroscopy is found to be a good probe for local ordering, in particular being sensitive to the first annealing steps, when the macroscopic order parameter is still high. Even though Raman data are scattered and there are outliers in the estimated values of the degree of order, the average peak linewidths of the Na-feldspar characteristic doublet band, labelled here as υa and υb, as a function of the order parameter Qod show interesting trends: both peak linewidths linearly increase until saturation. From Qod values lower than 0.6, peak broadening is no more affected by the Al-Si distribution. Moreover, the disordering process is found to be heterogeneous. SC-XRD and Raman data have suggested an inter-crystalline inhomogeneity of the samples, i.e., the presence of regions with different defect density on the micrometric scale. Finally, the influence of Ca-Na substitution in the plagioclase Raman spectra has been investigated. Raman spectra have been collected on a series of well characterized natural, low structural plagioclases. The variations of the Raman modes as a function of the chemical composition and the structural order have been determined. The number of the observed Raman bands at each composition gives information about the unit-cell symmetry: moving away from the C1 structures, the number of the Raman bands enhances, as the number of formula units in the unit cell increases. The modification from an “albite-like” Raman spectrum to a more “anorthite-like” spectrum occurs from sample An78 onwards, which coincides with the appearance of c reflections in the diffraction patterns of the samples. The evolution of the Raman bands υa and υb displays two changes in slope at ~An45 and ~An75: the first one occurs between e2 and e1 plagioclases, the latter separates e1 and I1 plagioclases with only b reflections in their diffraction patterns from I1 and P1 samples having b and c reflections too. The first variation represents exactly the e2→e1 phase transitions, whereas the second one corresponds in good approximation to the C1→I1 transition, which has been determined at ~An70 by previous works. The I1→P1 phase transition in the anorthite-rich side of the solid solution is not highlighted in the collected Raman spectra. Variations in peak broadening provide insights into the behaviour of the order parameter on a local scale, suggesting an increase in the structural disorder within the solid solution, as the structures have to incorporate more Al atoms to balance the change from monovalent to divalent cations. All the information acquired on these natural plagioclases has been used to produce a protocol able to give a preliminary estimation of the chemical composition of an unknown plagioclase from its Raman spectrum. Two calibration curves, one for albite-rich plagioclases and the other one for the anorthite-rich plagioclases, have been proposed by relating the peak linewidth of the most intense Raman band υa and the An content. It has been pointed out that the dependence of the composition from the linewidth can be obtained only for low structural plagioclases with a degree of order not far away from the references. The proposed tool has been tested on three mineralogical samples, two of meteoric origin and one of volcanic origin. Chemical compositions by Raman spectroscopy compare well, within an error of about 10%, with those obtained by elemental techniques. Further analyses on plagioclases with unknown composition will be necessary to validate the suggested method and introduce it as routine tool for the determination of the chemical composition from Raman data in planetary missions.

Geometric Hamiltonian formulation of a variational problem depending on the covariant acceleration

We consider a second-order variational problem depending on the covariant acceleration, which is related to the notion of Riemannian cubic polynomials. This problem and the corresponding optimal control problem are described in the context of higher order tangent bundles using geometric tools. The main tool, a presymplectic variant of Pontryagin’s maximum principle, allows us to study the dynamics of the control problem.