Coherent states and wave packet dynamics for the Bogoliubov-de Gennes equations - Langham-Lopez, Jordan (2016)
We investigate generalizations of coherent states as a means of representing the dynamics of excitations of the superconducting ground state. We also analyse the propagation of generalized coherent state wave packets under the Bogoliubov-de Gennes Hamiltonian. The excitations of the superconducting ground state are superpositions of electron and hole quasi-particles described by the Bogoliubov-de Gennes equations, that can only exist at energies outside the band gap. A natural generalization relevant to the excitations of the superconducting ground state is the tensor product of canonical and spin coherent states. This state will quickly become de-localized on phase space under evolution by the Bogoliubov-de Gennes Hamiltonian due to the opposite velocities of the quasi-spin components. We therefore define the electron-hole coherent states which remain localised on phase space over longer times. We show that the electron-hole coherent states though entangled retain many defining features of coherent states. We analyse the propagation of both product and electron hole coherent states in a superconductor with a spatially homogeneous superconducting band gap. The dispersion relation indicates that wavepackets defined on the band gap have a zero group velocity, but we will show that interference effects can create states on the band gap that propagate at the Fermi velocity. We also consider the two semiclassical, short wavelength regimes, hbar->0$ and the large Fermi energy limit mu->infinity. In general these limits produce behaviour analogous to the canonical coherent states except for isolated cases. Finally we analyse the dynamics of the Andreev Reflection of a Gaussian wavepacket incident on a discontinuous normal-superconducting interface. We show that restricting the energy bandwidth of the incident state inside the superconducting band gap precludes the wavepacket from fully entering the superconducting region. We again consider the two semiclassical regimes.