Apparent Superluminal Speeds in Evanescent Fields, Quantum Tunnelling and Quantum Entanglement

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Figure 2a. A rectangular glass prism has the property that a light ray from, e g, a LED to the left entering a side face of the prism can only exit through the opposite side face, where it is being detected (or through the same side face after reflections). No matter how many successive reflections may take place in a rectangular prism, the light ray will always experience total internal reflection in the other side faces. Similarly, external light will always be totally reflected in the side faces and cannot reach the detector. Using optical glass with a slightly higher refractive index, this separation between the ray systems can be maintained even in water (rain will not disturb the operation)

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Figure 2b. However, at total internal reflection an evanescent field (red, range exaggerated in the figure) extends into the outer medium and can be absorbed there by a finger. This thus causes part of the totally reflected light to vanish and not reach the detector. In Figure 1 the prism was attached in optical contact with the window, but since all surfaces are rectangular, the same separation between the orthogonal ray systems was maintained, i e external light will not interfere with the operation

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In one application, the optical touch key was used on shop windows, on the inside of which the keyboard was semi-permanently glued in optical contact, as shown in Figure 1. Displays in the shop window could in this way be activated by touching the outside of the window (Figures 2a and 2b explain the role of the evanescent field in this case in somewhat more detail).

Total internal reflection of this kind has also been demonstrated with microwaves in a block of paraffin, and where the range of the evanescent field outside the prism then can be of the order of centimetres.

When studying total internal reflection of the type occurring in the optical touch key described above, it seemed difficult at the time to understand the behaviour of a photon that had just experienced total internal reflection at the surface inside the glass (or paraffin), and had thus now moved some distance inside the glass back away from the surface. How could – from a pedestrian point of view – such a photon retrace its path after a while and instead decide to disappear just because someone had in the meantime put a finger within reach of the evanescent field on the outside of the surface and at some distance away from it? It seemed difficult to understand this in any other way than that the evanescent field outside the glass or paraffin must have propagated with infinite speed. Only in that way would it seem to have got the information of a possible outside breach some distance away of the total internal reflection and been able to convey this information to the photon already when it was at the surface, and thus allow the reflection not to occur.

It is interesting to see how diffuse, unspecified clutter of thoughts like this can precede new insights or discoveries, just like, e g, a similar case of the clutter of thoughts relating to dark energy/matter before it was known to exist at all ^{[5, 6, 7]}. Or, in the words of Jorge Luis Borges ^[8]: “the word ‘precursor’ ... should be cleansed of all connotations of polemic or rivalry. The fact is that every writer creates his own precursors. His work modifies our conception of the past, as it will modify the future.”

The main factor in the progress in the case of superluminal velocities has been the advent of methods to measure extremely short time intervals in the laboratory. This has now thus finally made it possible, notably in the experiments by Steinberg, Kwiat, and Chiao ^[9], to study such possible superluminal velocities experimentally. In these experiments a tunnelled photon was measured to have propagated with a velocity 1.7 times the speed of light. This will be further discussed in Sec. 3 below.

The existence of a possible superluminal component at total internal reflection has been described in earlier texts, but then historically considered not to be very relevant but mainly to be a relic of a non-quantum description. However, since the effect occurs also in large systems such as in microwave cavities ^[10] or a large block of paraffin, it needs to be taken seriously. Whatever effects Maxwell’s equations predict for this kind of behaviour must thus be understood to be real and not to be artefacts just due to the use of Maxwell’s equations on microscopic systems outside their range of validity.

Detailed theoretical calculations by Barbero, Hernández-Figueroa, and Recami ^[10], of evanescent fields in undersized waveguides show (cf Figure 3a and Figure 3b) the onset of the electromagnetic field to propagate with the speed of light, and then first to give a tiny Sommerfeld precursor followed by a larger Brillouin precursor before the main wave arrives. In this main wave, group velocities may very well exceed the speed of light, but will never be able to overtake the start of the wave. However, superluminal group velocities of this kind are not considered to be able to propagate information, which would thus exclude, e g, superluminal communication.

3. Superluminal Speed at Quantum Tunnelling?

Chiao, Kwiat, and Steinberg have further discussed their results mentioned above in an article ^[11] in Scientific American. In this article, the authors explain their results by picturing two tortoises competing to reach the other end of a field, each tortoise representing the wave packet of a particle in their experiment as illustrated in Figure 4.

One of the tortoises has to pass through a barrier, where it splits into two: one small tortoise that passes through the barrier whereas the rest of the tortoise gets reflected. If the tortoises crawl with the same speed, the nose of the tortoise that had no obstacle and the nose of the small tortoise will both pass the finishing line at the same time.

But the point is that if we detect the arrival of the tortoises at the finishing line not by their noses (i e the somewhat hard-to-measure front of the minute Sommerfeld precursor at t₀ in Figure 3b) but instead by the top of their backs (i e the maximum of the wave packets as in Figure 3c), then the smaller tortoise will seem to be ahead as illustrated in Figure 4.

Interpreted in this way, there would thus no longer be any real superluminal speed involved in this experiment – nor in the related optical touch key discussed in Sec. 2 – only a superluminal group velocity. Since the wave packet corresponding to the smaller tortoise has shrunk in length due to its superluminal group velocity, then as a consequence its maximum has thus been shifted forwards.

Figure 3a. A propagating wave packet in an undersized waveguide (from Barbero, Hernández-Figueroa and Recami [10] (reprinted with permission from assocpub@aps.org)

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Figure 3b. The same wave packet as in Figure 3a with the same time scale but with the amplitude scale enlarged by a factor of 10⁴, showing the tiny Sommerfeld precursor at t₀ preceding the somewhat larger Brillouin precursor at t₁ before the bulk of the wave packet arrives [10] (reprinted with permission from assocpub@aps.org)

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Figure 3c. Comparison between wave packets with different group velocities. The group velocity in the smaller, red wave packet is higher than in the main wave packet. Despite the precursors for both wave packets in this case arriving at the same time, the maximum of the red wave packet arrives before the maximum of the main wave packet due to the compression caused by its higher group velocity. This thus appears as a superluminal velocity as is also illustrated in Figure 4

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Figure 4. Superluminal group velocity illustrated [11] by two tortoises competing to reach the other end of a field. A barrier splits one of the tortoises into two: one part being reflected by the barrier and the other part being transmitted through it. Even though the tortoises both crawl with the same speed c, and their noses thus reach the finishing line at the same time, the top of the back of the smaller tortoise is ahead compared to the top of the back of the bigger tortoise, illustrating how group velocities in a barrier experiment thus can exceed the velocity c (reprinted with permission from Scientific American and Patricia J. Wynne, Wynneart@aol.com)

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4. Superluminal Speed at Quantum Entanglement?

As apparent from Einstein’s comment cited above of ‘spukhafte Fernwirkung’, quantum entanglement has been a controversial subject for a long time.

Experiments showing quantum entanglement have recently been performed with photons, electrons, molecules including large molecules like buckyballs ^[12] (C₆₀), and even small diamonds ^[13]. Of particular interest in connection to the question of superluminal speeds are experiments showing entanglement between quanta separated by large distances ^[14], and where thus some effects must be interpreted to have propagated with superluminal speed. This would thus seem to be in conflict with basic principles of special relativity.

However, concerning such seemingly superluminal velocities in connection with entangled photons, or other entangled particles moving essentially at light speed, these superluminal velocities can be understood as a relativistic effect. This will be discussed in the following, and it seems especially enlightening to discuss this question in relation to the assumed clock-hypothesis in special relativity.

The clock-hypothesis has comparatively recently ^[15] been introduced into special relativity as a new postulate, and states that accelerated motion of a system does not affect the passage of time in that system compared to a corresponding non-accelerating system. The clock-hypothesis now seems to be essentially generally accepted ^[16] as a universal principle in special relativity, and claiming to prohibit time to be dependent on acceleration in all situations.

Figure 5. Minkowski diagram showing how the space and time axes in moving systems (primed) relate to the corresponding axes in a stationary system xτ. Spacetime points (x_i, 0) transform as hyperbolas (red) for increasing relativistic velocities, and which coincide with the hyperbolic spacetime trajectories for particles in constant proper acceleration. Time shown by a clock moving along such a hyperbolic spacetime trajectory will thus appear slowed-down or even frozen to a stationary observer. Light is the special limiting case when x_i = 0 (blue). Then all points (x, x) of the light ray in the stationary system coalesce into one point (0, 0) in the co-moving system – thus making quantum entanglement understandable even between entangled photons separated by large distances in the stationary system as illustrated in Figure 6

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However, a problem is then that the clock hypothesis would seem to be in conflict with acceleration-gravitation equivalence, since gravitation is well documented to affect the passage of time, most spectacularly in black holes – and accelerations should thus be expected to do so too in corresponding cases. Furthermore, such a principle requiring time always to be independent of accelerations would obviously also require time in a system in hyperbolic spacetime motion to be unaffected by this particular type of acceleration and thus just tick on as it would if there were no acceleration.

Figure 6. Schematic illustration of how two entangled photons in the stationary system are observed in their own system to be simultaneously at the same point as discussed in Figure 5. Their entire wave functions in the stationary system transform into a point in their own system, which explains how measurement of the state of one of them is instantaneously transferred to the other one

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