248 research outputs found
Dynamics and Synchronization of Weak Chimera States for a Coupled Oscillator System
This thesis is an investigation of chimera states in a network of identical coupled phase
oscillators. Chimera states are intriguing phenomena that can occur in systems of coupled
identical phase oscillators when synchronized and desynchronized oscillators coexist.
We use the Kuramoto model and coupling function of Hansel for a specific system of six
oscillators to prove the existence of chimera states.
More precisely, we prove analytically there are chimera states in a small network of
six phase oscillators previously investigated numerically by Ashwin and Burylko [8]. We
can reduce to a two-dimensional system within an invariant subspace, in terms of phase
differences. This system is found to have an integral of motion for a specific choice of
parameters. Using this we prove there is a set of periodic orbits that is a weak chimera.
Moreover, we are able to confirmthat there is an infinite number of chimera states at the
special case of parameters, using the weak chimera definition of [8].
We approximate the Poincaré return map for these weak chimera solutions and demonstrate
several results about their stability and bifurcation for nearby parameters. These agree
with numerical path following of the solutions.
We also consider another invariant subspace to reduce the Kuramoto model of six
coupled phase oscillators to a first order differential equation. We analyse this equation
numerically and find regions of attracting chimera states exist within this invariant subspace.
By computing eigenvalues at a nonhyperbolic point for the system of phase differences, we
numerically find there are chimera states in the invariant subspace that are attracting within
full system.Republic of Iraq,
Ministry of Higher Education and Scientific Research
Threshold Switching and Self-Oscillation in Niobium Oxide
Volatile threshold switching, or current controlled negative
differential resistance (CC-NDR), has been observed in a range of
transition metal oxides. Threshold switching devices exhibit a
large non-linear change in electrical conductivity, switching
from an insulating to a metallic state under external stimuli.
Compact, scalable and low power threshold switching devices are
of significant interest for use in existing and emerging
technologies, including as a selector element in high-density
memory arrays and as solid-state oscillators for hardware-based
neuromorphic computing.
This thesis explores the threshold switching in amorphous NbOx
and the properties of individual and coupled oscillators based on
this response. The study begins with an investigation of
threshold switching in Pt/NbOx/TiN devices as a function device
area, NbOx film thickness and temperature, which provides
important insight into the structure of the self-assembled
switching region. The devices exhibit combined threshold-memory
behaviour after an initial voltage-controlled forming
process, but exhibit symmetric threshold switching when the RESET
and SET currents are kept below a critical value. In this mode,
the threshold and hold voltages are shown to be independent of
the device area and film thickness, and the threshold power,
while independent of device area, is shown to decrease with
increasing film thickness. These results are shown to be
consistent with a structure in which the threshold switching
volume is confined, both laterally and vertically, to the region
between the residual memory filament and the electrode, and where
the memory filament has a core-shell structure comprising a
metallic core and a semiconducting shell. The veracity of this
structure is demonstrated by comparing experimental results with
the predictions of a resistor network model, and detailed finite
element simulations.
The next study focuses on electrical self-oscillation of an NbOx
threshold switching device incorporated into a Pearson-Anson
circuit configuration. Measurements confirm stable operation of
the oscillator at source voltages as low as 1.06 V, and
demonstrate frequency control in the range from 2.5 to 20.5 MHz
with maximum frequency tuning range of 18 MHz/V. The oscillator
exhibit three distinct oscillation regimes: sporadic spiking,
stable oscillation and damped oscillation. The oscillation
frequency, peak-to-peak amplitude and frequency are shown to be
temperature and voltage dependent with stable oscillation
achieved for temperatures up to ∼380 K. A physics-based
threshold switching model with inclusion of device and circuit
parameters is shown to explain the oscillation waveform and
characteristic.
The final study explores the oscillation dynamics of capacitively
coupled Nb/Nb2O5 relaxation oscillators. The coupled system
exhibits rich collective behaviour, from weak coupling to
synchronisation, depending on the negative differential
resistance response of the individual devices, the operating
voltage and the coupling capacitance. These coupled oscillators
are shown to exhibit stable frequency and phase locking states at
source voltages as low as 2.2 V with MHz frequency tunable range.
The numerical simulation of the coupled system highlights the
role of source voltage, and circuit and device capacitance in
controlling the coupling modes and dynamics
Quantum computing using native interaction in superconducting circuits
Superconducting circuits form one of the most promising hardware platforms for building a quantum computer. As the quantum computing system gets more complex as we increase the size, employing simple circuit designs and control strategies can make the task of building a large scale quantum computer easier.
This thesis describes a novel control strategy that utilises spin-echo techniques and native interaction in superconducting circuits, which reduces the cost of calibrating pulsed two-qubit gates. Spin-echo pulses are used to rescale the always-on Hamiltonian, and the timings of spin-echo pulses encode the effective coupling strengths. In collaboration with the NMR group in Oxford, two methods for scaling this technique to large numbers of qubits were explored. In the first approach, pulse sequences for an all-to-all coupled system are obtained numerically using linear programming, and it finds the time-optimal solution for up to twenty qubits and the near time-optimal solution for up to hundreds of qubits. Another approach based on graph colouring finds the near time-optimal pulse sequence analytically, allowing pulse sequences for any number of qubits. An idea based on the Hamiltonian rescaling technique was applied to implementing the variational quantum eigensolver algorithm and error mitigation on two superconducting qubits. In contrast to previous studies, the residual dispersive coupling between qubits was used for computation instead of regarding it as a source of error.
Lastly, the detailed dynamics of the residual dispersive coupling in superconducting circuits were investigated to predict the practicality of spin-echo-based quantum computing on superconducting circuits. The Hamiltonian rescaling protocol assumes the always-on coupling to be diagonal, such as Ising Hamiltonian, but deviation from the pure Ising interaction was observed in the strongly coupled superconducting qubits. The origin of the deviation was identified analytically, and the circuit design criteria to suppress the deviation are presented
Quantum complex networks
This Thesis focuses on networks of interacting quantum harmonic oscillators and in particular, on them as environments for an open quantum system, their probing via the open system, their transport properties, and their experimental implementation. Exact Gaussian dynamics of such networks is considered throughout the Thesis.
Networks of interacting quantum systems have been used to model structured environments before, but most studies have considered either small or non-complex networks. Here this problem is addressed by investigating what kind of environments complex networks of quantum systems are, with specific attention paid on the presence or absence of memory effects (non-Markovianity) of the reduced open system dynamics. The probing of complex networks is considered in two different scenarios: when the probe can be coupled to any system in the network, and when it can be coupled to just one. It is shown that for identical oscillators and uniform interaction strengths between them, much can be said about the network also in the latter case. The problem of discriminating between two networks is also discussed.
While state transfer between two sites in a (typically non-complex) network is a well-known problem, this Thesis considers a more general setting where multiple parties send and receive quantum information simultaneously through a quantum network. It is discussed what properties would make a network suited for efficient routing, and what is needed for a systematic search and ranking of such networks. Finding such networks complex enough to be resilient to random node or link failures would be ideal.
The merit and applicability of the work described so far depends crucially on the ability to implement networks of both reasonable size and complex structure, which is something the previous proposals lack. The ability to implement several different networks with a fixed experimental setup is also highly desirable. In this Thesis the problem is solved with a proposal of a fully reconfigurable experimental realization, based on mapping the network dynamics to a multimode optical platform
SU(N) Fermions in a One-Dimensional Harmonic Trap
We conduct a theoretical study of SU(N) fermions confined by a
one-dimensional harmonic potential. Firstly, we introduce a new numerical
approach for solving the trapped interacting few-body problem, by which one may
obtain accurate energy spectra across the full range of interaction strengths.
In the strong-coupling limit, we map the SU(N) Hamiltonian to a spin-chain
model. We then show that an existing, extremely accurate ansatz - derived for a
Heisenberg SU(2) spin chain - is extendable to these N-component systems.
Lastly, we consider balanced SU(N) Fermi gases that have an equal number of
particles in each spin state for N=2, 3, 4. In the weak- and strong-coupling
regimes, we find that the ground-state energies rapidly converge to their
expected values in the thermodynamic limit with increasing atom number. This
suggests that the many-body energetics of N-component fermions may be
accurately inferred from the corresponding few-body systems of N
distinguishable particles.Comment: 15 pages, 6 figure
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Unconventional computing platforms and nature-inspired methods for solving hard optimisation problems
The search for novel hardware beyond the traditional von Neumann architecture has given rise to a modern area of unconventional computing requiring the efforts of mathematicians, physicists and engineers. Many analogue physical systems, including networks of nonlinear oscillators, lasers, condensates, and superconducting qubits, are proposed and realised to address challenging computational problems from various areas of social and physical sciences and technology. Understanding the underlying physical process by which the system finds the solutions to such problems often leads to new optimisation algorithms. This thesis focuses on studying gain-dissipative systems and nature-inspired algorithms that form a hybrid architecture that may soon rival classical hardware.
Chapter 1 lays the necessary foundation and explains various interdisciplinary terms that are used throughout the dissertation. In particular, connections between the optimisation problems and spin Hamiltonians are established, their computational complexity classes are explained, and the most prominent physical platforms for spin Hamiltonian implementation are reviewed.
Chapter 2 demonstrates a large variety of behaviours encapsulated in networks of polariton condensates, which are a vivid example of a gain-dissipative system we use throughout the thesis. We explain how the variations of experimentally tunable parameters allow the networks of polariton condensates to represent different oscillator models. We derive analytic expressions for the interactions between two spatially separated polariton condensates and show various synchronisation regimes for periodic chains of condensates. An odd number of condensates at the vertices of a regular polygon leads to a spontaneous formation of a giant multiply-quantised vortex at the centre of a polygon. Numerical simulations of all studied configurations of polariton condensates are performed with a mean-field approach with some theoretically proposed physical phenomena supported by the relevant experiments.
Chapter 3 examines the potential of polariton graphs to find the low-energy minima of the spin Hamiltonians. By associating a spin with a condensate phase, the minima of the XY model are achieved for simple configurations of spatially-interacting polariton condensates. We argue that such implementation of gain-dissipative simulators limits their applicability to the classes of easily solvable problems since the parameters of a particular Hamiltonian depend on the node occupancies that are not known a priori. To overcome this difficulty, we propose to adjust pumping intensities and coupling strengths dynamically. We further theoretically suggest how the discrete Ising and -state planar Potts models with or without external fields can be simulated using gain-dissipative platforms. The underlying operational principle originates from a combination of resonant and non-resonant pumping. Spatial anisotropy of pump and dissipation profiles enables an effective control of the sign and intensity of the coupling strength between any two neighbouring sites, which we demonstrate with a two dimensional square lattice of polariton condensates. For an accurate minimisation of discrete and continuous spin Hamiltonians, we propose a fully controllable polaritonic XY-Ising machine based on a network of geometrically isolated polariton condensates.
In Chapter 4, we look at classical computing rivals and study nature-inspired methods for optimising spin Hamiltonians. Based on the operational principles of gain-dissipative machines, we develop a novel class of gain-dissipative algorithms for the optimisation of discrete and continuous problems and show its performance in comparison with traditional optimisation techniques. Besides looking at traditional heuristic methods for Ising minimisation, such as the Hopfield-Tank neural networks and parallel tempering, we consider a recent physics-inspired algorithm, namely chaotic amplitude control, and exact commercial solver, Gurobi. For a proper evaluation of physical simulators, we further discuss the importance of detecting easy instances of hard combinatorial optimisation problems. The Ising model for certain interaction matrices, that are commonly used for evaluating the performance of unconventional computing machines and assumed to be exponentially hard, is shown to be solvable in polynomial time including the Mobius ladder graphs and Mattis spin glasses.
In Chapter 5 we discuss possible future applications of unconventional computing platforms including emulation of search algorithms such as PageRank, realisation of a proof-of-work protocol for blockchain technology, and reservoir computing
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