C. A. Holmes

General description of quasiadiabatic dynamical phenomena near exceptional points

The appearance of so-called exceptional points in the complex spectra of non-Hermitian systems is often associated with phenomena that contradict our physical intuition. One example of particular interest is the state-exchange process predicted for an adiabatic encircling of an exceptional point. In this work we analyze this and related processes for the generic system of two coupled oscillator modes with loss or gain. We identify a characteristic system evolution consisting of periods of quasistationarity interrupted by abrupt nonadiabatic transitions and we present a qualitative and quantitative description of this switching behavior by connecting the problem to the phenomenon of stability loss delay. This approach makes accurate predictions for the breakdown of the adiabatic theorem as well as the occurrence of chiral behavior observed previously in this context and provides a general framework to model and understand quasiadiabatic dynamical effects in non-Hermitian systems.

Synchronization of many nanomechanical resonators coupled via a common cavity field

Using amplitude equations, we show that groups of identical nanomechanical resonators, interacting with a common mode of a cavity microwave field, synchronize to form a single mechanical mode which couples to the cavity with a strength dependent on the squared sum of the individual mechanical-microwave couplings. Classically this system is dominated by periodic behavior which, when analyzed using amplitude equations, can be shown to exhibit multistability. In contrast, groups of sufficiently dissimilar nanomechanical oscillators may lose synchronization and oscillate out of phase at significantly higher amplitudes. Further, the method by which synchronization is lost resembles that for large amplitude forcing which is not of the Kuramoto form.

Critical fluctuations in cortical models near instability.

Computational studies often proceed from the premise that cortical dynamics operate in a linearly stable domain, where fluctuations dissipate quickly and show only short memory. Studies of human electroencephalography (EEG), however, have shown significant autocorrelation at time lags on the scale of minutes, indicating the need to consider regimes where non-linearities influence the dynamics. Statistical properties such as increased autocorrelation length, increased variance, power law scaling, and bistable switching have been suggested as generic indicators of the approach to bifurcation in non-linear dynamical systems. We study temporal fluctuations in a widely-employed computational model (the Jansen–Rit model) of cortical activity, examining the statistical signatures that accompany bifurcations. Approaching supercritical Hopf bifurcations through tuning of the background excitatory input, we find a dramatic increase in the autocorrelation length that depends sensitively on the direction in phase space of the input fluctuations and hence on which neuronal subpopulation is stochastically perturbed. Similar dependence on the input direction is found in the distribution of fluctuation size and duration, which show power law scaling that extends over four orders of magnitude at the Hopf bifurcation. We conjecture that the alignment in phase space between the input noise vector and the center manifold of the Hopf bifurcation is directly linked to these changes. These results are consistent with the possibility of statistical indicators of linear instability being detectable in real EEG time series. However, even in a simple cortical model, we find that these indicators may not necessarily be visible even when bifurcations are present because their expression can depend sensitively on the neuronal pathway of incoming fluctuations.

Hamiltonian mappings and circle packing phase spaces

We introduce three area preserving maps with phase space structures which resemble circle packings. Each mapping is derived from a kicked Hamiltonian system with one of the three different phase space geometries (planar, hyperbolic or spherical) and exhibits an infinite number of coexisting stable periodic orbits which appear to ‘pack’ the phase space with circular resonances.

Quantum Dynamics of Three Coupled Atomic Bose-Einstein Condensates

The simplest model of three coupled Bose-Einstein condensates is investigated using a group theoretical method. The stationary solutions are determined using the SU(3) group under the mean-field approximation. This semiclassical analysis, using system symmetries, shows a transition in the dynamics of the system from self trapping to delocalization at a critical value for the coupling between the condensates. The global dynamics are investigated by examination of the stable points, and our analysis shows that the structure of the stable points depends on the ratio of the condensate coupling to the particle-particle interaction, and undergoes bifurcations as this ratio is varied. This semiclassical model is compared to a full quantum treatment, which also displays a dynamical transition. The quantum case has collapse and revival sequences superimposed on the semiclassical dynamics, reflecting the underlying discreteness of the spectrum. Nonzero circular current states are also demonstrated as one of the higher-dimensional effects displayed in this system.

Bounded solutions of the nonlinear parabolic amplitude equation for plane Poiseuille flow

The stability conditions of plane waves against three-dimensional perturbations in plane Poiseuille flow, as described by a dispersive cubically nonlinear complex-amplitude equation, under perturbations quasi-periodic in two of the space dimensions are investigated. It is found that if the parameters satisfy certain conditions, a wave is totally stable. These conditions are an extension of those given for the lower dimensional case by J. T. Stuart and R. C. DiPrima (Proc R. Soc. Lond. A 362, 27-41 (1978)). The centre manifold theorem is then used to investigate the nature of the solutions bifurcating from a marginally unstable plane wave. Hopf bifurcations occur in the 1, 2 or 3 perturbing sidebands that are neutrally stable to the unperturbed wave and can give rise to limit cycles or tori.

Dynamics of a strongly driven two-component Bose-Einstein condensate

We consider a two-component Bose-Einstein condensate in two spatially localized modes of a double-well potential, with periodic modulation of the tunnel coupling between the two modes. We treat the driven quantum field using a two-mode expansion and define the quantum dynamics in terms of the Floquet Operator for the time periodic Hamiltonian of the system. It has been shown that the corresponding semiclassical mean-field dynamics can exhibit regions of regular and chaotic motion. We show here that the quantum dynamics can exhibit dynamical tunneling between regions of regular motion, centered on fixed points (resonances) of the semiclassical dynamics.

Quantum noise in the electromechanical shuttle: Quantum master equation treatment

We consider a type of quantum electromechanical system, known as the shuttle system, first proposed by Gorelik [Phys. Rev. Lett. 80, 4526 (1998)]. We use a quantum master equation treatment and compare the semiclassical solution to a full quantum simulation to reveal the dynamics, followed by a discussion of the current noise of the system. The transition between tunneling and shuttling regime can be measured directly in the spectrum of the noise. (c) 2006 American Institute of Physics.

Quantum signatures of chaos in the dynamics of a trapped ion

We show how a nonlinear chaotic system, the parametrically kicked nonlinear oscillator, may be realised in the dynamics of a trapped, laser-cooled ion, interacting with a sequence of standing wave pulses. Unlike the original optical scheme [G.J. Milburn and C.A. Holmes, Phys. Rev A, 44, p4704, (1991)], the trapped ion enables strongly quantum dynamics with minimal dissipation. This should permit an experimental test of one of the quantum signatures of chaos; irregular collapse and revival dynamics of the average vibrational energy.

Induced transparency in optomechanically coupled resonators

In this work we theoretically investigate a hybrid system of two optomechanically coupled resonators, which exhibits induced transparency. This is realized by coupling an optical ring resonator to a toroid. In the semiclassical analyses, the system displays bistabilities, isolated branches (isolas), and self-sustained oscillation dynamics. Furthermore, we find that the induced transparency window sensitively relies on the mechanical motion. Based on this fact, we show that the described system can be used as a weak force detector and the optimal sensitivity can beat the standard quantum limit without using feedback control or squeezing under available experimental conditions.