Abstract:
Mesoscopic superpositions of distinguishable coherent states provide an analog to the Schr\"odinger's cat thought experiment. For mechanical oscillators these have primarily been realised using coherent wavepackets, for which the distinguishability arises due to the spatial separation of the superposed states. Here, we demonstrate superpositions composed of squeezed wavepackets, which we generate by applying an internal-state dependent force to a single trapped ion initialized in a squeezed vacuum state with 9 dB reduction in the quadrature variance. This allows us to characterise the initial squeezed wavepacket by monitoring the onset of spin-motion entanglement, and to verify the evolution of the number states of the oscillator as a function of the duration of the force. In both cases, we observe clear differences between displacements aligned with the squeezed and anti-squeezed axes. We observe coherent revivals when inverting the state-dependent force after separating the wavepackets by more than 19 times the ground-state root mean squared extent, which corresponds to 56 times the root mean squared extent of the squeezed wavepacket along the displacement direction. Aside from their fundamental nature, these states may be useful for quantum metrology or quantum information processing with continuous variables.

Abstract:
The time-development of photoexcitations in molecular aggregates exhibits specific dynamics of electronic states and vibrational wavefunction. We discuss the dynamical formation of entanglement between electronic and vibrational degrees of freedom in molecular aggregates with theory of electronic energy transfer and the method of vibronic 2D wavepackets [Cina, Kilin, Humble, J. Chem. Phys. 118, 46 (2003)]. The vibronic dynamics is also described by applying Jaynes-Cummings model to the electronic energy transfer [Kilin, Pereverzev, Prezhdo, J. Chem. Phys. 120, 11209 (2004);math-ph/0403023]. Following the ultrafast excitation of donor[chem-ph/9411004] the population of acceptor rises by small portions per each vibrational period, oscillates force and back between donor and acceptor with later damping and partial revivals of this oscillation. The transfer rate gets larger as donor wavepacket approaches the acceptor equilibrium configuration, which is possible at specific energy differences of donor and acceptor and at maximal amount of the vibrational motion along the line that links donor and acceptor equilibria positions. The four-pulse phase-locked nonlinear wavepacket 2D interferograms reflect the shape of the relevant 2D vibronic wavepackets and have maxima at longer delay between excitation pulses for dimers with equal donor-acceptor energy difference compare to dimers with activationless energy configuration [Cina, Fleming, J. Phys. Chem. A. 108, 11196 (2004)].

Abstract:
We analyze the semiclassical evolution of Gaussian wavepackets in chaotic systems. We prove that after some short time a Gaussian wavepacket becomes a primitive WKB state. From then on, the state can be propagated using the standard TDWKB scheme. Complex trajectories are not necessary to account for the long-time propagation. The Wigner function of the evolving state develops the structure of a classical filament plus quantum oscillations, with phase and amplitude being determined by geometric properties of a classical manifold.

Abstract:
This work continues our studies of nonlinear evolution of a system of wavepackets. We study a wave propagation governed by a nonlinear system of hyperbolic PDE's with constant coefficients with the initial data being a multi-wavepacket. By definition a general wavepacket has a well defined principal wave vector, and, as we proved in previous works, the nonlinear dynamics preserves systems of wavepackets and their principal wave vectors. Here we study the nonlinear evolution of a special class of wavepackets, namely particle-like wavepackets. A particle-like wavepacket is of a dual nature: on one hand, it is a wave with a well defined principal wave vector, on the other hand, it a particle in the sense that it can be assigned a well defined position in the space. We prove that under the nonlinear evolution a generic multi-particle wavepacket remains to be a multi-particle wavepacket with a high accuracy, and every constituting single particle-like wavepacket not only preserves its principal wave number but also it has a well-defined space position evolving with a constant velocity which is its group velocity. Remarkably the described properties hold though the involved single particle-like wavepackets undergo nonlinear interactions and multiple collisions in the space. We also prove that if principal wavevectors of multi-particle wavepacket are generic, the result of nonlinear interactions between different wavepackets is small and the approximate linear superposition principle holds uniformly with respect to the initial spatial positions of wavepackets.

Abstract:
We study nonlinear systems of hyperbolic (in a wider sense) PDE's in entire d-dimensional space describing wave propagation with the initial data in the form of a finite sum of wavepackets referred to as multi-wavepackets. The problem involves two small parameters beta and rho where: (i) (1/beta) is a factor describing spatial extension of the wavepackets; (ii) (1/rho) is a factor describing the relative magnitude of the linear part of the evolution equation compared to its nonlinearity. For a wide range of the small parameters and on time intervals long enough for strong nonlinear effects we prove that multi-wavepackets are preserved under the nonlinear evolution. In particular, the corresponding wave vectors and the band numbers of involved wavepackets are "conserved quantities". We also prove that the evolution of a multi-wavepacket is described with high accuracy by a properly constructed system of envelope equations with a universal nonlinearity which in simpler cases turn into well-known Nonlinear Schrodinger or coupled modes equations. The universal nonlinearity is obtained by a certain time averaging applied to the original nonlinearity. This can be viewed as an extension of the well known averaging method developed for finite-dimensional nonlinear oscillatory systems to the case of a general translation invariant PDE systems with the linear part having continuous spectrum.

Abstract:
The experimental realization of lattices with Chern bands in ultracold-atom and photonic systems has motivated the study of time-dependent phenomena, such as spatial propagation, in lattices with nontrivial topology. We study the dynamics of gaussian wavepackets on the Haldane honeycomb Chern-band lattice model, in the presence of a harmonic trap. We focus on the transverse response to a force, which is due partly to the Berry curvature and partly to the transverse component of the energy band curvature. We evaluate the accuracy of a semiclassical description, which treats the wavepacket as a point particle in both real and momentum space, in reproducing the motion of a realistic wavepacket with finite extent. We find that, in order to accurately capture the wavepacket dynamics, the extent of the wavepacket in momentum space needs to be taken into account. The dynamics is sensitive to the interplay of band dispersion and Berry curvature over the finite region of momentum (reciprocal) space where the wavepacket has support. Moreover, if the wavepacket is prepared with a finite initial momentum, the semiclassical analysis reproduces its motion as long as it has a large overlap with the eigenstates of a single band. The semiclassical description generally improves with increasing real-space size of the wavepacket, as long as the external conditions (e.g., external force) remain uniform throughout the spatial extent of the wavepacket.

Abstract:
We depict and analyze a new effect for wavepackets falling freely under a barrier or well. The effect appears for wavepackets whose initial spread is smaller than the combination $\ds \sqrt{\frac{l_g^3}{|z_0|}}$, between the gravitational length scale $\ds l_g = \frac{1}{(2 m^2 g)^{1/3}}$ and the initial location of the packet $z_0$. It consists of a diffractive structure that is generated by the falling and spreading wavepacket and the waves reflected from the obstacle. The effect is enhanced when the Gross-Pitaevskii interaction for positive scattering length is included. The theoretical analysis reproduces the essential features of the effect. Experiments emanating from the findings are proposed.

Abstract:
The fundamental dynamics of ultracold atomtronic devices are reflected in their phonon modes of excitation. We probe such a spectrum by applying a harmonically driven potential barrier to a $^{23}$Na Bose-Einstein condensate in a ring-shaped trap. This perturbation excites phonon wavepackets. When excited resonantly, these wavepackets display a regular periodic structure. The resonant frequencies depend upon the particular configuration of the barrier, but are commensurate with the orbital frequency of a Bogoliubov sound wave traveling around the ring. Energy transfer to the condensate over many cycles of the periodic wavepacket motion causes enhanced atom loss from the trap at resonant frequencies. Solutions of the time-dependent Gross-Pitaevskii equation exhibit quantitative agreement with the experimental data. We also observe the generation of supersonic shock waves under conditions of strong excitation, and collisions of two shock wavepackets.

Abstract:
We derive a general upper bound on the spreading rate of wavepackets in the framework of Schr\"odinger time evolution. Our result consists of showing that a portion of the wavepacket cannot escape outside a ball whose size grows dynamically in time, where the rate of this growth is determined by properties of the spectral measure and by spatial properties of solutions of an associated time independent Schr\"odinger equation. We also derive a new lower bound on the spreading rate, which is strongly connected with our upper bound. We apply these new bounds to the Fibonacci Hamiltonian--the most studied one-dimensional model of quasicrystals. As a result, we obtain for this model upper and lower dynamical bounds establishing wavepacket spreading rates which are intermediate between ballistic transport and localization. The bounds have the same qualitative behavior in the limit of large coupling.

Abstract:
Wavepackets in quantum mechanics spread and the Universe in cosmology expands. We discuss a formalism where the two effects can be unified. The basic assumption is that the Universe is determined by a unitarily evolving wavepacket defined on space-time. Space-time is static but the Universe is dynamic. Spreading analogous to expansion known from observational cosmology is obtained if one regards time evolution as a discrete process with probabilities of jumps determined by a variational principle employing Kolmogorov-Nagumo-R\'enyi averages. The choice of the R\'enyi calculus implies that the form of the Universe involves an implicit fractal structure. The formalism automatically leads to two types of "time" parameters: $\tau$, with dimension of $x^0$, and dimensionless $\varepsilon=\ln \epsilon_\tau$, related to the form of diffeomorphism that defines the dynamics. There is no preferred time foliation, but effectively the dynamics leads to asymptotic concentration of the Universe on spacelike surfaces that propagate in space-time. The analysis is performed explicitly in $1+1$ dimensions, but the unitary evolution operator is brought to a form that makes generalizations to other dimensions and other fields quite natural.