Research

Dynamics of Bose-Einstein condensates

Predicted in 1924, but first observed only in 1995, a Bose-Einstein condensate is an exotic form of matter in which a macroscopic number of particles occupies the same quantum state. Behaving as a coherent quantum object, BECs have fascinating dynamical properties, and we investigate how quantum interference effects can be used to manipulate them - for example, by controlling the quantum phase transition between the Mott state and the superfluid, manipulating the self-trapping transition, and precisely controlling the motion of the constituent atoms.

As well as being of theoretical interest, controlling BECs is also highly relevant to the construction of quantum communication channels and quantum gates - and may provide the route to the creation of quantum computers.

Quantum chaos and cold atoms

Although they lack the coherence properties of BECs, systems of cold, trapped atoms provide a fascinating window into the transition between the classical and quantum worlds. Cold atoms subjected to pulsed optical potentials provide an almost perfect realization of the quantum kicked rotor (the paradigmatic model of quantum chaos), and varying the kick strength and kicking-period allows us to control the effective size of Planck's constant. The standard kicked rotor exhibits an effect known as dynamical localization - the energy of the system initially rises linearly, following the classical behaviour, until quantum effects cause it to saturate. By employing a different kick-pattern - double-kicks - we have shown how classical phase space barriers can be created and used to control the flow of atoms. In this way one can realize a purely Hamiltonian ratchet (i.e. without dissipation).

Strong correlations and nanostructures

The properties of low dimensional systems can be greatly influenced by the presence of inter-particle interactions, typically provided by the Coulomb interaction. Possibly the most extreme example is in one-dimension, in which Landau theory breaks down, and confined electrons undergo spin-charge separation and are described in the Luttinger liquid picture. Both static and transport properties of electronic systems such as quantum wires and quantum dots can be substantially modified by interactions, producing effects such as Coulomb blockade and Wigner crystallization, which will become of increasing importance as the size of electronic devices shrinks to the quantum regime. Systems of coupled quantum dots can have excellent coherence properties, and so quantum interference effects can be used to control their dynamics. We have investigated the interplay between interactions and interference effects by applying time-dependent driving fields, and have shown how electronic states can be manipulated in this way.

Quantum Monte Carlo: simulation and analytical continuation

Systems which do not have exact solutions, or which are not amenable to perturbation theory, are often the most interesting. This includes most low-dimensional systems with strong interactions, notably the two-dimensional Hubbard model, thought to be relevant to the phenomenon of high-temperature superconductivity. Techniques such as exact diagonalization allow the properties of systems of a small number of particles to be evaluated exactly, but to treat larger systems Quantum Monte Carlo simulations really come into their own. In this way we have simulated the half-filled Hubbard model, spin-1/2 and spin-1 Heisenberg chains, strongly correlated electrons in quantum wires, the Holstein model, and the vortex dynamics of the two-dimensional Coulomb gas. A complication is that the simulations use "imaginary time" which represents the temperature of the system, and so to obtain real-time correlation functions, susceptibilities, etc it is necessary to analytically continue the imaginary-time correlators to the real axis. This is an extremely unstable process in general - related to the case of numerically inverting a Laplace transform - but by combining ideas from maximum analysis with singular value decomposition we have shown how this may be done to produce stable results of high-resolution.