Asymptotic theory of quasiperiodically driven quantum systems

The theoretical treatment of quasi-periodically driven quantum systems is complicated by the inapplicability of the Floquet theorem, which requires strict periodicity. In this work we consider a quantum system driven by a bi-harmonic driving and examine its asymptotic long-time limit, the limit in which features distinguishing systems with periodic and quasi-periodic driving occur. Also, in the classical case this limit is known to exhibit universal scaling, independent of the system details, with the system's reponse under quasi-periodic driving being described in terms of nearby periodically driven system results. We introduce a theoretical framework appropriate for the treatment of the quasi-periodically driven quantum system in the long-time limit, and derive an expression, based on Floquet states for a periodically driven system approximating the different steps of the time evolution, for the asymptotic scaling of relevant quantities for the system at hand. These expressions are tested numerically, finding excellent agreement for the finite-time average velocity in a prototypical quantum ratchet consisting of a space-symmetric potential and a time-asymmetric oscillating force

An infinite branching hierarchy of time-periodic solutions of the Benjamin-Ono equation

We present a new representation of solutions of the Benjamin-Ono equation that are periodic in space and time. Up to an additive constant and a Galilean transformation, each of these solutions is a previously known, multi-periodic solution; however, the new representation unifies the subset of such solutions with a fixed spatial period and a continuously varying temporal period into a single network of smooth manifolds connected together by an infinite hierarchy of bifurcations. Our representation explicitly describes the evolution of the Fourier modes of the solution as well as the particle trajectories in a meromorphic representation of these solutions; therefore, we have also solved the problem of finding periodic solutions of the ordinary differential equation governing these particles, including a description of a bifurcation mechanism for adding or removing particles without destroying periodicity. We illustrate the types of bifurcation that occur with several examples, including degenerate bifurcations not predicted by linearization about traveling waves.Comment: 27 pages, 6 figure

Momentum-space atom correlations in a Mott insulator

We report on the investigation of the three-dimensional single-atom-resolved distributions of bosonic Mott insulators in momentum-space. Firstly, we measure the two-body and three-body correlations deep in the Mott regime, finding a perfectly contrasted bunching whose periodicity reproduces the reciprocal lattice. In addition, we show that the two-body correlation length is inversely proportional to the in-trap size of the Mott state with a pre-factor in agreement with the prediction for an incoherent state occupying a uniformly filled lattice. Our findings indicate that the momentum-space correlations of a Mott insulator at small tunnelling is that of a many-body ground-state with Gaussian statistics. Secondly, in the Mott insulating regime with increasing tunnelling, we extract the spectral weight of the quasi-particles from the momentum density profiles. On approaching the transition towards a superfluid, the momentum spread of the spectral weight is found to decrease as a result of the increased mobility of the quasi-particles in the lattice. While the shapes of the observed spectral weight agree with the ones predicted by perturbative many-body calculations, the fitted mobilities are larger than the theoretical ones. This discrepancy is similar to that previously reported on the time-of-flight visibility.Comment: 13 pages, 10 figure

Applied Symmetry for Crystal Structure Prediction

This thesis presents an original open-source Python package called PyXtal (pronounced pi-crystal ) that generates random symmetric crystal structures for use in crystal structure prediction (CSP). The primary advantage of PyXtal over existing structure generation tools is its unique symmetrization method. For molecular structures, PyXtal uses an original algorithm to determine the compatibility of molecular point group symmetry with Wyckoff site symmetry. This allows the molecules in generated structures to occupy special Wyckoff positions without breaking the structure\u27s symmetry. This is a new feature which increases the space of search-able structures and in turn improves CSP performance. It is shown that using already-symmetric initial structures results in a higher probability of finding the lowest-energy structure. Ultimately, this lowers the computational time needed for CSP. Structures can be generated for any symmetry group of 0, 1, 2, or 3 dimensions of periodicity. Either atoms or rigid molecules may be used as building blocks. The generated structures can be optimized with VASP, LAMMPS, or other computational tools. Additional options are provided for the lattice and inter-atomic distance constraints. Results for carbon and silicon crystals, water ice crystals, and molybdenum clusters are presented as usage examples

Delay Parameter Selection in Permutation Entropy Using Topological Data Analysis

Permutation Entropy (PE) is a powerful tool for quantifying the predictability of a sequence which includes measuring the regularity of a time series. Despite its successful application in a variety of scientific domains, PE requires a judicious choice of the delay parameter $\tau$. While another parameter of interest in PE is the motif dimension $n$, Typically $n$ is selected between $4$ and $8$ with $5$ or $6$ giving optimal results for the majority of systems. Therefore, in this work we focus solely on choosing the delay parameter. Selecting $\tau$ is often accomplished using trial and error guided by the expertise of domain scientists. However, in this paper, we show that persistent homology, the flag ship tool from Topological Data Analysis (TDA) toolset, provides an approach for the automatic selection of $\tau$. We evaluate the successful identification of a suitable $\tau$ from our TDA-based approach by comparing our results to a variety of examples in published literature

Qubit state transfer via discrete-time quantum walks

We propose a scheme for perfect transfer of an unknown qubit state via the discrete-time quantum walk on a line or a circle. For this purpose, we introduce an additional coin operator which is applied at the end of the walk. This operator does not depend on the state to be transferred. We show that perfect state transfer over an arbitrary distance can be achieved only if the walk is driven by an identity or a flip coin operator. Other biased coin operators and Hadamard coin allow perfect state transfer over finite distances only. Furthermore, we show that quantum walks ending with a perfect state transfer are periodic.Comment: 13 pages, 5 figure

Geometric Phase, Hannay's Angle, and an Exact Action Variable

Canonical structure of a generalized time-periodic harmonic oscillator is studied by finding the exact action variable (invariant). Hannay's angle is defined if closed curves of constant action variables return to the same curves in phase space after a time evolution. The condition for the existence of Hannay's angle turns out to be identical to that for the existence of a complete set of (quasi)periodic wave functions. Hannay's angle is calculated, and it is shown that Berry's relation of semiclassical origin on geometric phase and Hannay's angle is exact for the cases considered.Comment: Submitted to Phys. Rev. Lett. (revised version