Low Energy Quantum System Simulation

A numerical method for solving Schrodinger's equation based upon a Baker-Campbell-Hausdorff (BCH) expansion of the time evolution operator is presented herein. The technique manifestly preserves wavefunction norm, and it can be applied to problems in any number of spatial dimensions. We also identify a particular dimensionless ratio of potential to kinetic energies as a key coupling constant. This coupling establishes characteristic length and time scales for a large class of low energy quantum states, and it guides the choice of step sizes in numerical work. Using the BCH method in conjunction with an imaginary time rotation, we compute low energy eigenstates for several quantum systems coupled to non-trivial background potentials. The approach is subsequently applied to the study of 1D propagating wave packets and 2D bound state time development. Failures of classical expectations uncovered by simulations of these simple systems help develop quantum intuition. Finally, we investigate the response of a Superconducting Quantum Interference Device (SQUID) to a time dependent potential. We discuss how to engineer the potential's energy and time scales so that the SQUID acts as a quantum NOT gate. The notional simulation we present for this gate provides useful insight into the design of one candidate building block for a quantum computer.