• stretching potential
• Wigner phase time
• Hartman effect
• quantum tunnelling
• tunnelling time
• epsilon-near-zero photonic metamaterial

The problem of the tunnelling time has been raised soon after the quantum tunnelling phenomenon surfaced in Hund’s work. Different characteristic time definitions have been proposed and many of them predict the Hartman effect, stating that the tunnelling time will saturate with increasing barrier length, resulting in an unbound tunnelling velocity. Many well-designed experiments verifying tunnelling with such superluminal velocities have been reported (in all cases without violating relativistic causality). Among different definitions, the Wigner phase time is well established and is of our interest in this thesis. Specifically, we investigate tunneling in a system with a ‘stretching potential’ whereby an electron wave function is flat on an extended spatially range, in analogy with the case of an epsilon-near-zero photonic metamaterial supporting waves with very large wavelengths at large frequencies. The phase time tunnelling through double barriers with a stretching potential region in the middle has been discussed in detail. The double barriers are set to be high, but finite, so that the confined state inside the stretching region becomes a quasi-bound state. By introducing a stretchable potential of increasing spatial range, the peculiar behavior of the resonant tunnelling has been observed and compared with that of traditional double barriers.