W-dispersion particles in repulsive potentials: Quasibound states and their lifetime

We consider fundamental features which emerge in the mechanics of quasiparticles with nonmonotonic (as a function of $p^2$) dispersion law. Quasiparticles of this kind abound in modern physics, with examples ranging from holes in quantum wells to edge magnetic states in quantum wires to photons in atomic vapors to polaritons in photonic crystals and in trapped-atom lattices. The motion of such a particle in repulsive potentials gives rise to a number of counterintuitive phenomena, which carry a promise of unusual optical manifestations. A classical particle can be trapped by repulsive potentials, and the likelihood of this trapping may increase with the value of the angular momentum. Further, in contrast to the usual quantum-mechanical notion, the particle always has a quasibound state in a two-dimensional, central-force repulsive potential, while it may have no bound states in a one-dimensional analog of this potential. The binding energy of these states and their inherent decay rate are determined by a complex interplay of the parameters of the potential, the particle dispersion law, and the value of the angular momentum. We construct the energy spectrum of quasibound states in a repulsive Coulomb potential, estimate their lifetime, and predict their optical manifestations as inverted hydrogen spectral-line series.

Figure 1

Left panel: Top: a typical surface of the effective Hamiltonian $H_M(p,\rho )$ in the case of free particle with $W$-shape-dispersion law. The coordinates are normalized with respect to the parameters of the dispersion law, $p_0$ and $\Delta$, and the value of the angular momentum, $M$. Bottom: the corresponding pattern of phase trajectories. Right panel: Top: a typical $H_M(p,\rho )$ surface for a particle in an axially symmetric repulsive potential. Bottom: the corresponding pattern of phase trajectories; closed phase trajectories signify the particle localization.