Nonadiabatic couplings and gauge-theoretical structure of curved quantum waveguides

We investigate the quantum mechanics of a single particle constrained to move along an arbitrary smooth reference curve by a confinement that is allowed to vary along the waveguide. The Schrödinger equation is evaluated in the adapted coordinate frame and a transverse-mode decomposition is performed, taking into account both curvature and torsion effects and the possibility of a cross-section potential that changes along the curve in an arbitrary way. We discuss the adiabatic structure of the problem, and examine nonadiabatic couplings that arise due to the curved geometry, the varying transverse profile, and their interplay. The exact multimode matrix Hamiltonian is taken as the natural starting point for few-mode approximations. Such approximate equations are provided, and it is worked out how these recover known results for twisting waveguides and can be applied to other types of waveguide designs. The quantum waveguide Hamiltonian is recast into a form that clearly illustrates how it generalizes the Born-Oppenheimer Hamiltonian encountered in molecular physics. In analogy to the latter, we explore the local gauge structure inherent to the quantum waveguide problem and suggest the usefulness of diabatic states, giving an explicit construction of the adiabatic-to-diabatic basis transformation.

Figure 1

(Left) An alternative type of quantum waveguide, obtained using a straight planar reference curve combined with a $u_1$-dependent displacement of the confining potential. This results in a repulsive effective potential proportional to $\dot{d}{\left(u_1\right)}^2$. (Right) A corresponding waveguide with the reference curve in its center and a $u_1$-independent confining potential perpendicular to it, resulting in an attractive effective potential proportional to ${\kappa }^2$ and independent of the transverse mode under consideration.