Tunneling Barriers by Günter Nimtz

II.Physikalisches Institut,Universität Köln, April 22,2004

Can light travel across space in zero time? Is there such a thing as negative energy? The strange world of quantum mechanical tunneling has the answers.

Tunneling, where a particle has a statistical chance of jumping straight through an apparently impassable barrier, is an essential component of physical processes like radioactivity and nuclear fusion. However, there is no experimental data on the time that the particle spends inside the barrier. How long do these strange jumps take?

Luckily, there is a related phenomenon that affects electromagnetic photons, whether in the form of visible light or microwaves. German physicist Hermann Ludwig von Helmholz (b 1821) derived a solution to the wave equation (effectively a sophisticated version of the statement that speed of wave is its wavelength times its frequency) that describes how electromagnetic waves propagate. The solution has both real and imaginary parts. (Imaginary in the mathematical sense, i.e. making use of the square root of minus 1.)

Traditionally these imaginary parts have been ignored as “imaginary numbers do not describe any real physical quantity”, but it was discovered that solutions that are purely imaginary (called evanescent modes), are mathematically identical with the solutions of the Schrödinger’s equation, the fundamental equation describing quantum mechanical behaviour in the case of tunneling [1].

Evanescent modes are not just a theoretical concept, but occur in a number of physical devices that are collectively called photonic barriers. These can, for instance be undersized waveguides. Waveguides are the rectangular cross-section metal tubes used to carry electromagnetic waves around the microwave part of the spectrum. In an undersized waveguide, a section of the waveguide that is smaller in cross-section than half the wavelength of the electromagnetic radiation (in both lateral directions) acts as a barrier.

Figure 1: Sketches of 3 photonic barriers.

a) Undersized waveguide b) A dielectric lattice c) Frustrated total internal reflection.

A second kind of barrier, a photonic lattice, is formed by a sandwich of layers of materials with very different refractive indices. These lattices stop photons in certain “forbidden” frequency bands passing through, except when they tunnel. The final example uses an air gap between double prisms to produce a condition called frustrated total reflection, a phenomenon hinted at by Newton. If two prisms are placed together with a small gap containing a less refracting substance (air, for instance), it is found that a small proportion of the light that should bounce back in the first prism when it is at angle that generates total internal reflection in fact “jumps the gap” tunneling through to the second prism and out.

Purely imaginary solutions of the wave equation seem to imply a zero shift in the phase of the wave – which would mean that the wave spent zero time in the barrier, crossing it instantaneously (or perhaps more accurately getting from one side of the barrier to the other without crossing the intervening space). With the electromagnetic analogies available, it was tempting to test tunneling time for real using photonic experiments.

Figure 2: Tunneling photons in frustrated total internal reflection

First microwave [3] and later optical experiments [4,5]were carried out to measure the total tunneling time of the photons.(In Fig.2: the tunneling time is tv+th) This strange, two part timing is due to the nature of frustrated total internal reflection. The photon is not a point, but extends out into the gap, so the reflection appears to take place behind the surface of the first prism, resulting in a shift down the surface before reflection, D in the diagram, called the Goos-Hänchen shift. This shift was conjectured by Newton 300 years ago, but only measured in 1947 by Goos and Hänchen.

The small value of the tunneling time results in velocities faster than light – photons crossing the barrier take less time than they should at light speed. In the actual experiment, using microwaves, it was found that both reflected and transmitted beams left their respective prisms at exactly the same time. With the distance d set at 60mm, the microwaves should have taken 20 picoseconds to cross the gap. However this time was not detected. The experiment was accurate to ±5 picoseconds, so something should have shown up. It seems that tunneling inside the barrier is nonlocal, proceeding in zero time. (Details of the experimental set-up are given in [6])

It might seem possible that the tunneling photon actually heads straight across the gap and doesn’t first undergo the shift – but the time taken is independent of the gap size and as near as can be identified identical with the time to cover the distance D.

Another interesting aspect of the tunneling process is that the predicted energy of the tunneling particles is negative. This fact, often overlooked in photonics, is noted in solid state physics [7], in quantum mechanics [8], and some textbooks on near-field optics. Are tunneling photons detectable? Theory says no, they should only be in evidence outside the barrier – see, for instance [8 and 9]. We have carried out a standard experiment with an undersized waveguide as shown in Fig.1a.

The undersized waveguide had a cut-off frequency of 9.49 GHz and the tunneling microwave had a frequency of 8.74 GHz. The transmission was extremely small, just -134 dB for the barrier length of 200 mm. To allow the pickup probe to penetrate inside the transmission to sample the electric field distribution, a tapered slot was cut in the undersized waveguide.

Photons with an energy above the barrier energy could easily be detected, but tunneling photons with lower energy could not be detected, although a number of photons had traversed the barrier and were measured after they had left the barrier. The tunnelling photons are not measureble inside the barrier due to their negative energy as predicted by quantum mechanics.

Detection only took place when the evanescent behaviour was destroyed by penetrating the pickup probe deep into the waveguide. In this case the originally purely imaginary impedance of the undersized waveguide obtained a real part and the barrier was annihilated. Tunneling was not necessary for traversing the narrow waveguide now and the reflection dropped by the amount of energy picked up by the probe. Tunneling particles (e.g. photons) have a negative energy inside the barrier – this prevents them from being measurable.

Tunneling displays a space of zero-time and a negative energy, both properties are describable by quantum mechanics.

Günter Nimtz is Professor of Physics at Köln University. He became well-known in the 1990s when undertaking experiments on superluminal velocities, where electromagnetic radiation exceeds the speed of light. An account of part of his work can be found in Brian Clegg’s Light Years.

Read more about Professor Nimtz’s work at his website

References

  1. R. Feynman, Lectures on Physics II 33 –12, Addison –Wesley (1970)
  2. G. Nimtz, Prog. Quantum Electronics, 417 (2003)
  3. A. Enders and G. Nimtz, J.Phys. I France 2 1693 (1992)
  4. A. M. Steinberg et al., Phys. Rev. Letters 71 708(1993)
  5. S. Longhi, M. Marano, P. Laporta, and M. Belmonte, Phys. Rev. E 64 055602 (2001)
  6. A. Haibel, G. Nimtz, and A. A. Stahlhofen, Phys. Rev. E 63 047601(2001)
  7. Y. Takada and W. Kohn, Phys. Rev. B 37 826 (1988)
  8. E. Merzbacher, Quantum Mechanics second Ed.p.91,J.Wiley &Sons, New York (1970)
  9. F. de Fornel, Evanescent waves, Springer series in optical sciences p.18,73 (2000)

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