Time-dependent massless Dirac fermions in graphene

Abstract

Using the Lewis-Riesenfeld method of invariants we construct explicit analytical solutions for the massless Dirac equation in 2+1 dimensions describing quasiparticles in graphene. The Hamiltonian of the system considered contains some explicit time-dependence in addition to one resulting from being minimally coupled to a timedependent vector potential. The eigenvalue equations for the two spinor components of the Lewis-Riesenfeld invariant are found to decouple into a pair of supersymmetric invariants in a similar fashion as the known decoupling for the time-independent Dirac Hamiltonians.The two dimensional massless Dirac equation has recently attracted a lot of renewed attention because it describes quasi-particles in graphene [1,2,3], which is well known to possess a large amount of remarkable properties. Especially the Dirac equation in the presence of a magnetic field is of great interest as, unlike electrostatic potentials, such a configuration allows in principle to confine the Dirac fermions [4,5,6]. Many exact solutions have been provided for a variety of time-independent Hamiltonians and magnetic field configurations [7,8,9], including some for complex magnetic fields leading to pseudo/quasi-Hermitian interactions [10,11]. While some solutions for the time-dependent Dirac equation in 1+1 dimensions have been constructed [12,13,14], little is known about the time-dependent setting with a magnetic field in 2+1 dimensions and no exact solutions have been reported. The aim of this manuscript is to commence filling that apparent gap. We shall demonstrate that the Lewis-Riesenfeld method of invariants [15] is a technique which can be employed successfully to solve this problem.We consider here the time-dependent massless Dirac equation in two spacial dimensions in the form HΨ = i∂ t Ψ, with H(x, y, t) = σ · p,and the two component wave function Ψ = (ψ 1 , ψ 2 ). The effective Hamiltonian includes σ = (σ x , σ y ) comprised of the standard Pauli matrices and p being the two-dimensional

Robust state preparation in quantum simulations of Dirac dynamics

“…For the Hamiltonian in Eq. ( 2), let us assume that a nontrivial invariant exists with the form [21][22][23]…”

“…Gaussian wave packet centered at zero momentum (green, solid line and blue, dot-dashed line for σ = 0.3 and σ = 0.9, respectively) and P 2 (red, dotted-dashed line and black circles for σ = 0.3 and σ = 0.9, respectively) by averaging over all momenta p 0 , see Eq. (23), during the population inversion. H 0 as in Fig.…”