Optimal unravellings for feedback control in linear quantum systems

For quantum systems with linear dynamics in phase space much of classical feedback control theory applies. However, there are some questions that are sensible only for the quantum case: Given a fixed interaction between the system and the environment what is the optimal measurement on the environment for a particular control problem? We show that for a broad class of optimal (state- based) control problems ( the stationary linear-quadratic-Gaussian class), this question is a semidefinite program. Moreover, the answer also applies to Markovian (current-based) feedback.

Quantum dynamics of a model for two Josephson-coupled Bose-Einstein condensates

In this work, we investigate the quantum dynamics of a model for two singlemode Bose-Einstein condensates which are coupled via Josephson tunnelling. Using direct numerical diagonalization of the Hamiltonian, we compute the time evolution of the expectation value for the relative particle number across a wide range of couplings. Our analysis shows that the system exhibits rich and complex behaviours varying between harmonic and non-harmonic oscillations, particularly around the threshold coupling between the delocalized and selftrapping phases. We show that these behaviours are dependent on both the initial state of the system and regime of the coupling. In addition, a study of the dynamics for the variance of the relative particle number expectation and the entanglement for different initial states is presented in detail.

Measurement-based teleportation along quantum spin chains

We examine the teleportation of an unknown spin-1/2 quantum state along a quantum spin chain with an even number of sites. Our protocol, using a sequence of Bell measurements, may be viewed as an iterated version of the 2-qubit protocol of C. H. Bennett et al. [Phys. Rev. Lett. 70, 1895 (1993)]. A decomposition of the Hilbert space of the spin chain into 4 vector spaces, called Bell subspaces, is given. It is established that any state from a Bell subspace may be used as a channel to perform unit fidelity teleportation. The space of all spin-0 many-body states, which includes the ground states of many known antiferromagnetic systems, belongs to a common Bell subspace. A channel-dependent teleportation parameter O is introduced, and a bound on the teleportation fidelity is given in terms of O.

A quantum yield map for synthetic eumelanin

The quantum yield of synthetic eumelanin is known to be extremely low and it has recently been reported to be dependent on excitation wavelength. In this paper, we present quantum yield as a function of excitation wavelength between 250 and 500 nm, showing it to be a factor of 4 higher at 250 nm than at 500 nm. In addition, we present a definitive map of the steady-state fluorescence as a function of excitation and emission wavelengths, and significantly, a three-dimensional map of the specific quantum yield: the fraction of photons absorbed at each wavelength that are subsequently radiated at each emission wavelength. This map contains clear features, which we attribute to certain structural models, and shows that radiative emission and specific quantum yield are negligible at emission wavelengths outside the range of 585 and 385 nm (2.2 and 3.2 eV), regardless of excitation wavelength. This information is important in the context of understanding melanin biofunctionality, and the quantum molecular biophysics therein. (c) 2005 American Institute of Physics.

Surface morphology dependent photoluminescence from colloidal silicon nanocrystals

Monodisperse 1-2 nm silicon nanocrystals are synthesized in reverse micelles and have their surfaces capped with either allylamine or 1-heptene to produce either hydrophilic or hydrophobic silicon nanocrystals. Optical characterization (absorption, PL, and time-resolved PL) is performed on colloidal solutions with the two types of surface-capped silicon nanocrystals with identical size distributions. Direct evidence is obtained for the modification of the optical properties of silicon nanocrystals by the surface-capping molecule. The two different surface-capped silicon nanocrystals show remarkably different optical properties.

First-principles quantum dynamics in interacting Bose gases II: stochastic gauges

First principles simulations of the quantum dynamics of interacting Bose gases using the stochastic gauge representation are analysed. In a companion paper, we showed how the positive-P representation can be applied to these problems using stochastic differential equations. That method, however, is limited by increased sampling error as time evolves. Here, we show how the sampling error can be greatly reduced and the simulation time significantly extended using stochastic gauges. In particular, local stochastic gauges (a subset) are investigated. Improvements are confirmed in numerical calculations of single-, double- and multi-mode systems in the weak-mode coupling regime. Convergence issues are investigated, including the recognition of two modes by which stochastic equations produced by phase-space methods in general can diverge: movable singularities and a noise-weight relationship. The example calculated here displays wave-like behaviour in spatial correlation functions propagating in a uniform 1D gas after a sudden change in the coupling constant. This could in principle be tested experimentally using Feshbach resonance methods.

On the distributed compression of quantum information

The problem of distributed compression for correlated quantum sources is considered. The classical version of this problem was solved by Slepian and Wolf, who showed that distributed compression could take full advantage of redundancy in the local sources created by the presence of correlations. Here it is shown that, in general, this is not the case for quantum sources, by proving a lower bound on the rate sum for irreducible sources of product states which is stronger than the one given by a naive application of Slepian-Wolf. Nonetheless, strategies taking advantage of correlation do exist for some special classes of quantum sources. For example, Devetak and Winter demonstrated the existence of such a strategy when one of the sources is classical. Optimal nontrivial strategies for a different extreme, sources of Bell states, are presented here. In addition, it is explained how distributed compression is connected to other problems in quantum information theory, including information-disturbance questions, entanglement distillation and quantum error correction.

Optimal control, geometry, and quantum computing

We prove upper and lower bounds relating the quantum gate complexity of a unitary operation, U, to the optimal control cost associated to the synthesis of U. These bounds apply for any optimal control problem, and can be used to show that the quantum gate complexity is essentially equivalent to the optimal control cost for a wide range of problems, including time-optimal control and finding minimal distances on certain Riemannian, sub-Riemannian, and Finslerian manifolds. These results generalize the results of [Nielsen, Dowling, Gu, and Doherty, Science 311, 1133 (2006)], which showed that the gate complexity can be related to distances on a Riemannian manifold.

Quantum computation as geometry

Quantum computers hold great promise for solving interesting computational problems, but it remains a challenge to find efficient quantum circuits that can perform these complicated tasks. Here we show that finding optimal quantum circuits is essentially equivalent to finding the shortest path between two points in a certain curved geometry. By recasting the problem of finding quantum circuits as a geometric problem, we open up the possibility of using the mathematical techniques of Riemannian geometry to suggest new quantum algorithms or to prove limitations on the power of quantum computers.

Spin-detection in a quantum electromechanical shuttle system

We study the electrical transport of a harmonically bound, single-molecule C-60 shuttle operating in the Coulomb blockade regime, i.e. single electron shuttling. In particular, we examine the dependance of the tunnel current on an ultra-small stationary force exerted on the shuttle. As an example, we consider the force exerted on an endohedral N@C-60 by the magnetic field gradient generated by a nearby nanomagnet. We derive a Hamiltonian for the full shuttle system which includes the metallic contacts, the spatially dependent tunnel couplings to the shuttle, the electronic and motional degrees of freedom of the shuttle itself and a coupling of the shuttle's motion to a phonon bath. We analyse the resulting quantum master equation and find that, due to the exponential dependence of the tunnel probability on the shuttle-contact separation, the current is highly sensitive to very small forces. In particular, we predict that the spin state of the endohedral electrons of N@C-60 in a large magnetic gradient field can be distinguished from the resulting current signals within a few tens of nanoseconds. This effect could prove useful for the detection of the endohedral spin-state of individual paramagnetic molecules such as N@C-60 and P@C-60, or the detection of very small static forces acting on a C-60 shuttle.

The exact sample complexity of PAC-learning problems with unit VC dimension

The Vapnik-Chervonenkis (VC) dimension is a combinatorial measure of a certain class of machine learning problems, which may be used to obtain upper and lower bounds on the number of training examples needed to learn to prescribed levels of accuracy. Most of the known bounds apply to the Probably Approximately Correct (PAC) framework, which is the framework within which we work in this paper. For a learning problem with some known VC dimension, much is known about the order of growth of the sample-size requirement of the problem, as a function of the PAC parameters. The exact value of sample-size requirement is however less well-known, and depends heavily on the particular learning algorithm being used. This is a major obstacle to the practical application of the VC dimension. Hence it is important to know exactly how the sample-size requirement depends on VC dimension, and with that in mind, we describe a general algorithm for learning problems having VC dimension 1. Its sample-size requirement is minimal (as a function of the PAC parameters), and turns out to be the same for all non-trivial learning problems having VC dimension 1. While the method used cannot be naively generalised to higher VC dimension, it suggests that optimal algorithm-dependent bounds may improve substantially on current upper bounds.

Regression with input-dependent noise: A Gaussian process treatment

Gaussian processes provide natural non-parametric prior distributions over regression functions. In this paper we consider regression problems where there is noise on the output, and the variance of the noise depends on the inputs. If we assume that the noise is a smooth function of the inputs, then it is natural to model the noise variance using a second Gaussian process, in addition to the Gaussian process governing the noise-free output value. We show that prior uncertainty about the parameters controlling both processes can be handled and that the posterior distribution of the noise rate can be sampled from using Markov chain Monte Carlo methods. Our results on a synthetic data set give a posterior noise variance that well-approximates the true variance.

Quantification of the relationship between, magnetoencephalographic (MEG) and blood oxygenation dependent (BOLD) images of brain function

Magnetoencephalography (MEG) is the measurement of the magnetic fields generated outside the head by the brain’s electrical activity. The technique offers the promise of high temporal and spatial resolution. There is however an ambiguity in the inversion process of estimating what goes on inside the head from what is measured outside. Other techniques, such as functional Magnetic Resonance Imaging (fMRI) have no such inversion problems yet suffer from poorer temporal resolution. In this study we examined metrics of mutual information and linear correlation between volumetric images from the two modalities. Measures of mutual information reveal a significant, non-linear, relationship between MEG and fMRI datasets across a number of frequency bands.

Synthesis and evaluation of novel ligands for quantum dot stabilisation and polymerisation studies

This thesis describes the design and synthesis of a variety of functionalised phosphine oxides and sulfides, based on the structure of trioctylphosphine oxide, synthesised for the purpose of surface modification of quantum dots. The ability of the ligands to modify the surface chemistry via displacement of the original hexadecylamine capping layer of quantum dots was evaluated. Finally the surface modified quantum dots were investigated for enhancement in their inherent properties and improved compatibility with the various applications for which they were initially designed. Upon the commencement of research involving quantum dots it became apparent that more information on their behaviour and interaction with the environment was required. The limits of the inherent stability of hexadecylamine capped quantum dots were investigated by exposure to a number of different environments. The effect upon the stability of the quantum dots was monitored by changes in the photoluminescence ability of their cores. Subtle differences between different batches of quantum dots were observed and the necessity to account for these in future applications noted. Lastly the displacement of the original hexadecylamine coating with the "designer" functionalised ligands was evaluated to produce a set of conditions that would result in the best possible surface modification. A general procedure was elucidated however it was discovered that each displacement still required slight adjustment by consideration of the other factors such as the difference in ligand structure and the individuality of the various batches of quantum dots. This thesis also describes a procedure for the addition of a protective layer to the surface of quantum dots by cross-linking the functionalised ligands bound to the surface via an acyclic diene metathesis polymerisation. A detailed description of the problems encountered in the analysis of these materials combined with the use of novel techniques such as diffusion ordered spectroscopy is provided as a means to overcome the limitations encountered. Finally a demonstration of the superior stability, upon exposure to a range of aggressive environments of these protected materials compared with those before cross-linking provided physical proof of the cross-linking process and the advantages of the cross-linking modification. Finally this thesis includes the presentation of initial work into the production of luminescent nanocrystal encoded resin beads for the specific use in solid phase combinatorial chemistry. Demonstration of the successful covalent incorporation of quantum dots into the polymeric matrices of non-functionalised and functionalised resin beads is described. Finally by preliminary work to address and overcome the possible limitations that may be encountered in the production and general employment of these materials in combinatorial techniques is given.

A procedure for determining a spacewise dependent heat source and the initial temperature

The inverse problem of determining a spacewise dependent heat source, together with the initial temperature for the parabolic heat equation, using the usual conditions of the direct problem and information from two supplementary temperature measurements at different instants of time is studied. These spacewise dependent temperature measurements ensure that this inverse problem has a unique solution, despite the solution being unstable, hence the problem is ill-posed. We propose an iterative algorithm for the stable reconstruction of both the initial data and the source based on a sequence of well-posed direct problems for the parabolic heat equation, which are solved at each iteration step using the boundary element method. The instability is overcome by stopping the iterations at the first iteration for which the discrepancy principle is satisfied. Numerical results are presented for a typical benchmark test example, which has the input measured data perturbed by increasing amounts of random noise. The numerical results show that the proposed procedure gives accurate numerical approximations in relatively few iterations.