Tunneling delay times in one and two dimensions

Phys. Rev. A 49, 3283 – Published 1 May 1994
Aephraim M. Steinberg and Raymond Y. Chiao


We demonstrate that although the well-known analogy between the time-independent solutions for two-dimensional tunneling (e.g., frustrated total internal reflection) and tunneling through a one-dimensional potential barrier cannot, in general, be extended to the time domain, there are certain limits in which the delay times for the two problems obey a simple relationship. In particular, when an effective mass is chosen such that mc2=ħω, the ‘‘classical’’ traversal times for allowed transmission become identical for a photon of energy ħω traversing an air gap between regions of index n and for a particle of mass m traversing the analogous square barrier of height V0 in one dimension. The quantum-mechanical group delays are also identical, given this effective mass, both for EV0 (θ≊θc) and for EV0 (θ≪θc). (For a smoothly varying potential or index of refraction, the agreement persists for all values of E where the WKB approximation applies.) 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, EV0 (θ≊π/2), one must choose mc2=n2ħω in order to make the group delays equal for the two problems. These equivalences simplify certain calculations, and the two-dimensional analogy may also be useful for geometrically visualizing the tunneling process and the anomalously small group delays known to occur in the opaque limit. We also demonstrate that the equality of the group delays for transmission and reflection for lossless barriers follows from a simple intuitive argument based on time-reversal invariance, and discuss the extension of the result to the case of lossy barriers.

DOI: http://dx.doi.org/10.1103/PhysRevA.49.3283

  • Received 18 May 1993
  • Published in the issue dated May 1994

© 1994 The American Physical Society

Authors & Affiliations

Aephraim M. Steinberg and Raymond Y. Chiao

  • Department of Physics, University of California at Berkeley, Berkeley, California 94720

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