Paul Strange

Quantum entanglement allows for correlations between distant objects that go beyond any classical theory. These additional correlations can be exploited to gain practical advantage in certain non-local games. In recent years there has been interest in games defined on graph structures and involving mobile agents. One of these is the graph domination game, where quantum advantage has been discovered recently on some finite graphs by numerical optimization [1]. Here we study quantum advantage in the 1-step, 2-player version of this game, focusing on cycle graphs. We study it numerically, analytically and through the use of noisy, intermediate scale quantum (NISQ) processors. We find explicit strategies and show that they realise the numerical bounds that were found recently for the case of small graphs [1]. We then generalise our strategies to cycles of arbitrary size. Finally, we run our strategies for 5-, 6-, and 7-site cycles on NISQ hardware and find measurable advantage (compared to the optimal classical strategies) in all cases.

We study quantum advantage in one-step rendezvous games on simple graphs analytically, numerically, and using noisy intermediate-scale quantum (NISQ) processors. Our protocols realise the recently discovered (Mironowicz 2023 New J. Phys. 25 013023) optimal bounds for small cycle graphs and cubic graphs. In the case of cycle graphs, we generalise the protocols to arbitrary graph size. The NISQ processor experiments realise the expected quantum advantage with high accuracy for rendezvous on the complete graph K ₃. In contrast, for the graph 2 K 4 , formed by two disconnected 4-vertex complete graphs, the performance of the NISQ hardware is sub-classical, consistent with the deeper circuit and known qubit decoherence and gate error rates.

Quantum backflow is the unexpected effect that wavepackets consisting of only positive momentum components can apparently move in the negative direction. This is usually described in terms of the backflow constant, which is a dimensionless quantity describing least upper bound on the amount of probability that can flow backwards during a given time interval. Backflow is usually calculated for wavepackets that can be written as a sum of positive momentum plane waves. Here we present a calculation of the backflow constant using the localised free particle hermite wavefunctions where equal weights of positive and negative momentum eigenfunctions occur. The resulting backflow constant is substantially smaller than the accepted value. The reasons for this are discussed and finally we draw conclusions about the calculation of backflow more generally.

If a wave function is written in polar form it becomes possible to write the Schrödinger equation of nonrelativistic quantum mechanics in a form analogous to the classical Hamilton-Jacobi equation with an extra term known as the quantum potential. Time-dependent supersymmetry is a procedure for finding new solutions of the Schrödinger equation if one solution is known. In this paper a time-dependent supersymmetry transformation is applied to a wave function in this polar form and it is shown that the classical potential plus the quantum potential is a conserved quantity under this transformation under certain circumstances. This leads to a modification of our view of the role of the quantum potential and also to a deeper appreciation of the function of a supersymmetry transformation.

We apply the time-dependent supersymmetry methods of Bagrov, Samsonov and Shekoyan to the Schrödinger equation for a quantum bouncer. A new family of potentials, that correspond to the linear gravitational potential with a small oscillatory component superimposed upon it, is produced. Both the frequency and amplitude of the oscillatory part can be controlled and the corresponding eigenfunctions are found. These solutions are explored and basic observables are calculated. In particular we examine how motion in such a potential can be distinguished from motion in a linear gravitational field. We also point out some pedagogical aspects of this project.

We analyse the evolution of superoscillations in a relativistic wavepacket. A simple superoscillating wavepacket is set up and is allowed to evolve freely according to both the Klein–Gordon equation and the Dirac equation. The superoscillations evolve anisotropically and decay after a time. Both the lifetime and anisotropy can be understood in terms of the interaction of contributions to the wavepacket from components with strongly differing complex wavenumber. The analysis is supported by numerical calculations and the results are compared with the non-relativistic analysis. A potential experiment in which the significance of relativistic effects on superoscillations could be measured is proposed.

Here we report a project in which time-dependent supersymmetry has been employed to derive a new potential and eigenfunctions that satisfy the Schrödinger equation. The supersymmetry method is outlined and we apply it to a wavefunction obeying the free-particle Schrödinger equation. This leads to an exactly soluble model in which a quantum particle is seen to ‘surf’ on a time-dependent potential. The model can be solved and understood within both classical and quantum mechanics and the relationship between the two approaches is discussed. The mathematics of this formalism is accessible to a final year British undergraduate making supersymmetry derived Hamiltonians suitable as a final year theoretical physics research project.

In this paper we discuss relativistic quantum backflow. The general theory of relativistic backflow is presented and it is shown that the backflow can be written as a function of a simple parameter ϵ, which is defined in terms of fundamental constants and the backflow period. Backflow eigenfunctions are determined numerically for a range of values of ϵ, and an explicit expression for the relativistic backflow eigenvalue in terms of the non-relativistic backflow constant is presented. Backflow eigenvectors are then fitted with some standard functions, which lead to substantially higher backflow than has been found previously with fitting procedures, for some values of ϵ . In analysing the non-relativistic limit of the theory we show that this problem is one of those rare cases where the relativistic theory is intrinsically more simple than the non-relativistic theory.

Lipkin’s zilches are a set of little-known conserved quantities in classical electromagnetic theory. Here we report a systematic calculation of the zilches for topologically non-trivial vacuum electromagnetic fields and their interpretation in terms of both the physical and mathematical properties of the fields. Several families of electromagnetic fields have been explored and examined computationally. In these cases it is found that the zilches can be written in terms of more familiar conserved quantities: energy, momentum and angular momentum. Furthermore we demonstrate that the zilches also contain information about the topology of the field lines for the fields we have examined, thus providing a previously unsuspected aspect to their interpretation. We conjecture that these properties generalise to all integrable fields.

In this paper we discuss quantum curl forces. We present both the classical and quantum theory of linear curl forces. The quantum theory is shown to reproduce the classical theory precisely if appropriate combinations of eigenfunctions are chosen. A series of examples are used to illustrate the theory and to demonstrate its limitations. Furthermore, we are able to point out an analogy between the quantum theory of curl forces and some of the squeezed light states of quantum optics.