Session J6: Nonlinear Dynamics

Ballistic Atom Pumps

Researchers have long been interested in electron transport through mesojunctions containing time-dependent potential barriers, a process often called ``quantum pumping.'' A useful model of such a system is a ballistic atom pump: two reservoirs of neutral ultracold atoms connected by a channel which has oscillating repulsive potential-energy barriers. Particles move through the pump independently, and only interact with the walls and potentials. Such a system can transport particles from one reservoir to the other, even when the reservoirs have equal chemical potentials. It can also transfer energy from one reservoir to the other, even if there is no net particle pumping. Another type of pump, a rectifier--which only allows current to flow in one direction--can be constructed by tuning the potentials. While these phenomena are often called ``quantum pumping,'' we have found that the quantum description cannot be fully understood without analysis of the underlying classical dynamics. Classically, the system is a nice model of chaotic transport. We use classical trajectories, along with phase information, to construct a semiclassical approximation to the quantum description. This approach explains the locations and relative heights of Floquet peaks seen in the quantum theory.   

Measuring the Geometric Phase of the Driven Harmonic Oscillator

Since Aharonov and Anandan discovered that, in general, quantum states acquire a geometric phases during evolution, regardless of whether the associated Hamiltonian evolves adiabatically or not; geometric phases have become present in the formulation of most fundamental physical statements. Here, we propose a system allowing direct observation of the geometric phase, arising after cyclic evolution, in one of the most fundamental physical systems, the driven quantum harmonic oscillator (DQHO). In our work we calculate the geometric phase for a general time-dependent DQHO and propose a scheme to measure it. The DQHO Hamiltonian can be realized using engineered arrays of coupled optical waveguides. In such an array the electric field amplitude exhibits the same dynamics as the wave function of a DQHO in the Fock base representation. Moreover, we show that the dynamic phase acquired by the electric field representing the oscillator state and the field traversing an isolated waveguide are the same. Hence, by superimposing both fields at the revival point, the emergent interference pattern will be governed only by the associated geometric phase. On this platform we can easily establish the exact dependence of the geometric phase on the parameters of the DQHO.   

Preservation of the identities of multiple overlapping solitons through a collision with a strong integrability-breaking barrier

We show that when a strongly-coupled ``breather'' of the nonlinear Schr\"{o}dinger equation is scattered off of a nonperturbatively strong barrier, the solitons constituting the breather survive the collision. As the barrier height is lowered with impact speed held constant, at first all solitons are fully reflected; then suddenly the smallest soliton starts being fully transmitted; after the barrier is lowered some more, suddenly the next smallest soliton begins being fully transmitted as well, etc. Why the constituent solitons survive the integrability-breaking collision process is at present a mystery.   

Direct observation, study and control of molecular super rotors

Extremely fast rotating molecules whose rotational energy is comparable with or exceeds the molecular bond strength are known as ``super rotors''. It has been speculated that super rotors may exhibit a number of unique properties, yet only indirect evidence of these molecular objects has been reported to date. We demonstrate the first direct observation of molecular super rotors by detecting coherent unidirectional molecular rotation with extreme frequencies exceeding 10 THz. The technique of an ``optical centrifuge'' is used to control the degree of rotational excitation in an ultra-broad range of rotational quantum numbers, reaching as high as N = 95 in oxygen and N = 60 in nitrogen. State-resolved detection enables us to determine the shape of the excited rotational wave packet and quantify the effect of centrifugal distortion on the rotational spectrum. Femtosecond time resolution reveals coherent rotational dynamics with increasing coherence times at higher angular momentum. We demonstrate that molecular super rotors can be created and observed in dense samples under normal conditions where the effects of ultrafast rotation on many-body interactions, inter-molecular collisions and chemical reactions can be readily explored.   

Centrifugal Distortion Causes Anderson Localization in Laser Kicked Molecules

The periodically kicked 2D rotor is a textbook model in nonlinear dynamics. The classical kicked rotor can exhibit truly chaotic motion, whilst in the quantum regime this chaotic motion is suppressed by a mechanism similar to Anderson Localization. Up to now, these effects have been mainly observed in an atom optics analogue of a quantum rotor: cold atoms in a standing light wave. We demonstrate that common linear molecules (like N$_2$, O$_2$, CO$_2$, ...), kicked by a train of short linearly polarized laser pulses, can exhibit a new mechanism for dynamical Anderson Localization due to their non-rigidity. When the pulses are separated by the rotational revival time $t_{rev}=\pi\hbar/B$, the angular momentum $J$ grows ballistically (Quantum Resonance). We show that, due to the centrifugal distortion of fast spinning molecules, above some critical value $J=J_{cr}$ the Quantum Resonance is suppressed via the mechanism of Anderson Localization. This leads to a non-sinusoidal oscillation of the angular momentum distribution, which may be experimentally observed even at ambient conditions by using current techniques for laser molecular alignment.   

Effective Long-Range Interactions for Charged Particles in Confined Curved Dimensions

We explore the effective long-range interaction of charged particles confined to a curved low-dimensional manifold using the example of a helical geometry. Opposite to the Coulomb interaction in free space the confined particles experience a force which is oscillating with the distance between the particles. This leads to stable equilibrium configurations and correspondingly induced bound states whose number is tunable with the parameters of the helix. We demonstrate the existence of a plethora of equilibria of few-body chains with different symmetry character that are allowed to freely move. An outline concerning the implications on many-body helical chains is provided. We explore the effects arising due to the coupling of the center of mass and relative motion of two charged particles confined on an inhomogeneous helix with a locally modified radius. For an inhomogeneous helix, the coupling of the center of mass and relative motion induces an energy transfer between the collective and relative motion, leading to dissociation ofinitially bound states in a scattering process. Due to the time reversal symmetry, a binding of the particles out of the scattering continuum is thus equally possible. We identify the regimes of dissociation for different initial conditions.   

The One-Dimensional Soft-Coulomb Problem and the Hard-Coulomb Limit

A new and efficient way of evolving a solution to an ordinary differential equation is presented. A finite element method is used where we expand in a convenient local basis set of functions that enforce both function and first derivative continuity across the boundary. We also, for the first time, implement an adaptive step size choice for each element that is based on a Taylor series expansion. This algorithm is used to solve for the eigenpairs corresponding to the one--dimensional soft Coulomb potential, $1/\sqrt{x^2 + \beta^2}$, which becomes numerically intractable as the softening parameter ($\beta$) approaches zero. We are able to maintain near machine accuracy for $\beta$ as low as $\beta = 10^{-8}$ using 16 digit precision calculations. Our numerical results provide a new insight into the controversial one dimensional Hydrogen atom which is a limiting case of the soft Coulomb problem as $\beta \rightarrow 0$.   

Quantum response to classical autoresonance

A classical nonlinear oscillator can be driven to increasingly higher energy by chirping the driving frequency with a linear chirp rate chosen by various protocols, including one involving the behavior of the Teager-Keizer energy operator. We explore the auto-resonance response to linear chirping in a quantum mechanical system. Firstly, we report on the effect of applying this protocol to quantum system, particularly as the system size is changed so that the effective Planck's constant increases in size and the behavior becomes more quantum-mechanical. In addition, we discuss the systems behavior as we vary the linear chirp rate, and relate the optimal linear chirping frequencies and the system's size. We also comment on how the system reacts when introducing a noise term to such a nonlinear oscillator. Lastly, we show the transient behaviour from auto-resonance to quantum ladder climbing as the system gets more quantum-mechanical.   

Using quantum fidelity to measure quantum chaotic behavior with the delta-kicked rotor

The quantum delta-kicked rotor has been one of the workhorses in the experimental investigation of quantum chaos. Most experiments have been performed using ultra-cold atoms that are exposed to a spatially and temporally periodic optical potential. The measurement of the atomic momentum distribution after such an interaction has taken place can allow for the observation of many interesting phenomena such as quantum resonances, dynamical localization, ratchets, and accelerator modes. Nevertheless, there are other aspects of this system that remain difficult to study through the momentum distribution alone. For example, understanding what makes a quantum system ``chaotic'' in the classical sense of exponentially diverging trajectories in phase space is an open question. In this presentation we show how quantum fidelity, defined as the overlap between two quantum states, may be used to address this problem. We will present results showing how the fidelity between states that have undergone different interactions with the optical potential depends on both the strength of the interactions and the temporal period of the potential. We also show how this can be used to infer information about the underlying quantum chaotic phase space.   

Coherent perfect rotation theory: connections with, and consequences beyond, the anti-laser

Coherent Perfect Rotation (CPR) phenomena are a reversible generalization of the anti-laser. By evaluating CPR in a broad variety of common optical systems, including optical cavities and DFB and DBR structures, we illustrate its unique threshold and resonance features. This study builds intuition critical to assessing the utility of CPR in optical devices, and we detail it in a concrete application.