TUNNELING DELAY TIMES IN ONE-DIMENSION AND 2-DIMENSIONS

We demonstrate that although the well-known analogy between the time-i ndependent solutions for two-dimensional tunneling (e.g., frustrated t otal internal reflection) and tunneling through a one-dimensional pote ntial barrier cannot, in general, be extended to the time domain, ther e are certain limits in which the delay times for the two problems obe y a simple relationship. In particular, when an effective mass is chos en such that mc2=homegaBAR, the ''classical'' traversal times for allo wed transmission become identical for a photon of energy homegaBAR tra versing an air gap between regions of index n and for a particle of ma ss m traversing the analogous square barrier of height V0 in one dimen sion. The quantum-mechanical group delays are also identical, given th is effective mass, both for E almost-equal-to V0 (theta almost-equal-t o theta(c)) and for E much greater than V0 (theta much less than theta (c)). (For a smoothly varying potential or index of refraction, the ag reement persists for all values of E where the WKB approximation appli es.) The same relation serves to equate the quantum-mechanical ''dwell '' times for any values of E and V0. On the other hand, in the ''deep tunneling'' limit, E much less than V0 (theta almost-equal-to pi/2), o ne must choose mc2 = n2homegaBAR in order to make the group delays equ al for the two problems. These equivalences simplify certain calculati ons, and the two-dimensional analogy may also be useful for geometrica lly visualizing the tunneling process and the anomalously small group delays known to occur in the opaque limit. We also demonstrate that th e equality of the group delays for transmission and reflection for los sless barriers follows from a simple intuitive argument based on time- reversal invariance, and discuss the extension of the result to the ca se of lossy barriers.