Puzzles and Mysteries of Physics in Everyday Life. Part 2.

Adventures in Quantum Mechanics

with the assistance of Larry Weinberg

Electromagnetic waves are characterized by their frequency ν, and their propagation velocity c which relates wavelength λ to frequency ν = c/λ. You will see these symbols often in this document so it’s good to get used to them.

Viewed from a fixed location it vibrates at a frequency inversely proportional to its wavelength.

To emphasize the connection between circular motion and wave motion, as you contemplate the figures, the generated circularly polarized wave propagates along the normal to the plane of the circle. Two linearly polarized waves generated by the oscillatory components of the circular motion will propagate along the coordinate axes. The wave polarized along x propagates along y, whereas the wave polarized along y propagates along x.

The wave nature of light was suspected by scientists through the 18th century but positively confirmed by the double slit interference experiment of Thomas Young in 1801. The basic interference demonstration setup is described next, in modern day language, mixing wave and particle (photons) aspects of light. As we learned more recently, electrons have wave properties as well, that are reflected in analogous experiments.

A small point-like light source sprays photons in many directions, in the form of a spherical wave. Ahead we interpose a flat screen with two parallel vertical slits. Some of the photons move right through the openings. These photons do not contribute to interference, only to a background signal. Those that hit the edges will be scattered in many directions, in the form now of a “cylindrical wave”.

Sufficiently narrow slits, only scatter radiation from the edges, as a pair of expanding cylindrical waves. The wave that started at the common point source is split into two waves as if emanating from the slits. That is the case for sufficiently narrow slits so that only a minimal fraction of the light can flow directly through. A light sensitive detector plane is used to sense for the light distribution beyond the slits. It could be a photographic plate, a video camera, a scanning photon counting detector, or nowadays, a photon counting detector array.

Interference effects will then show up in the image on the detecting plane as vertical illuminated streaks, parallel to the slits. (Figure 9) Two light beams originating simultaneously from the same source point but scattered by the edges of both slits can superpose on the detecting plane thereby displaying the interference effects. The difference in the path lengths as it relates to the wavelength determines the strength of the registered signal. Superposition “in phase” doubles the strength. In opposition cancels it to zero. For a given slit separation the measured geometry of this interference pattern will yield the value of the wavelength of the light.

Bottom is in phase.
Top is almost in opposition.

An almost-plane wave is split into two cylindrical waves emanating from the slits. On the detection plane the interference fringes appear. The plot on the right shows the corresponding power density distribution.

In order to obtain an exactly parallel beam via lenses and mirrors, no matter how perfect they are, one must start with an exact point source with no spatial extension. That is the reason for the qualifier “almost”. A laser source might come close. This issue will reoccur in the discussion of phase coherence.

Note that this plot is always positive whereas the amplitude of the wave oscillates. This is the basic property of waves. The amplitude of a wave oscillates symmetrically between equal positive and negative extremes. Thus, over multiple cycles its average value vanishes. However, we know that the effects of waves are cumulative. Ocean waves caused serious damage in Fukushima.

Radiation collected by solar panels charge up the lithium ion batteries that power the Tesla. So how do we obtain this average power? Simply by squaring the amplitude which is always positive and by taking the average over many cycles. The wave detector whether it is a photographic plate, a photodiode or a photon counter does the squaring and the averaging. When you square the superposition of two identical waves with differing phases basic trigonometry predicts the interference pattern.

The signal is strongest at the center where the two waves come together in phase as implied by the equality of the lengths of the two paths. However, if they superpose off center the path lengths will differ and if this difference is an odd multiple of half a wavelength they will cancel. If the path difference happens to equal any integer multiple of the full wavelength the two waves will superpose in phase, the signal strength doubles but as the detection pixel moves progressively away from the center the detected signal will weaken as the constant pixel detection area subtends a progressively narrower cone of the arriving radiation.

In order to see interference fringes, both slits must be open simultaneously. When we record the signal with one slit
blocked there are no interference fringes, as shown in the top of figures 12(A) and 13(B). With both slits open we obtain the pictures at the bottom, showing the interference fringes.

Figure 13(B) show interference fringes as recorded by a photon counting array.

The first display shows no fringes as the signal was the accumulation of signals from two sequential exposures with alternating slits open one at a time. The next 3 display fringes recorded with the detector array rotating away from being parallel to the plane of the slits. Again, no interference unless both slits are simultaneously open.

This puzzling effect, just by itself, illuminates the dual nature of radiation. The power that is the energy flow carried by the radiation is not the whole story.

The entity that represents radiation is a time dependent field, that carries, besides a magnitude, a phase signature imposed at the instant of first observation. It turns out that the power carried by the radiation is related to its field magnitude by a quadratic relation. When the superposition is applied to the field rather than to the powers the interference shows up. This brings up the notion of phase coherence between radiation signals detected by separate detectors. At this point we assume that the notion of phase of a vibration, or of a wave, is now understood.

Consider a point light source behind the double slit screen. If on the detector screen we see the interference fringes we label the source coherent and the degree of coherence is the ratio of the intensity of the fringes to the intensity of the background. Perfect coherence requires identical single frequency sources. The best approximation to a coherent source would be a laser focused by a lens through a pinhole.

By observing sunlight light coming off Newton’s prism through a narrow slit suitably located, a narrow range of colors, therefore wavelengths or frequencies, can be isolated from the continuum emitted by the sun or by an incandescent light bulb. As shown in the figure the same could be achieved by suitable variants of double slit interference. A laser will do much better. This is because the laser generates light by “Light Amplification by Stimulated Emission of Radiation”. This process was identified by Einstein in 1905 as a consequence of the laws of electromagnetism. A discussion of this process is left for the section on lasers.

In the next paragraphs of this section we illustrate the hidden process that makes the transition between photons and waves. Furthermore, we discuss the idea that the concept of phase can be preserved through this transition.

Figure 14 below illustrates the buildup of interference from the random but correlated arrival of photons from a laser source to a photon counting array detector. The sequence a, b, c, d. corresponds to steps of increasing light intensity. Each dot indicates the arrival of one photon.

This figure shows how we move from the particle aspect to the wave aspect of radiation. The wave nature manifests itself as a coherence among the photons of common origin that is preserved during propagation.

It is as if each photon carries a clock running at a constant frequency determined by its energy. All photons of the same energy have their clocks running at the same frequency. The only difference can be the phase difference which is determined by the difference in starting times. The clock starts to tic at the instant of its creation, when a particle makes a downward transition between two energy levels. It stops when the photon disappears, giving up its energy to a particle, enabling it to make an upwards transition.

If, shortly after, the same particle makes the reverse transition, another identical photon is created but with a shifted phase. The phase shift is a direct measure of the time delay. This is the quantum definition of an elastic collision.

However, if the circumstances of the particle change during the delay, the energy, and therefore, the clock frequency will differ from that of the original incoming photon, and this is an inelastic collision, usually accompanied by energy absorption by the particle.

For small frequency changes and for short times this looks or feels just like a phase change, but this process starts to wash out their capability to participate in the interference process. When a phase change becomes a frequency change, photons are being replaced by new photons in rapid steps and the frequency spectrum broadens. This is what we call decoherence.

A perfectly elastic collision does not cause decoherence. The narrower the frequency bandwidth of a sharp light beam, the higher its degree of coherence provided all photons follow “equal” paths. Equal, here, means that every photon in the beam must undergo only perfectly elastic collisions. As the beam diverges, generally, inelastic collisions are taking place. However, if it is redirected by highly polished mirrors or lenses a degree of coherence can be recovered. The property of coherence, just as the property of polarization, manifests itself most obviously at sufficiently high intensity that the number density of photons is very large.

Adventures in Quantum Mechanics. Part 3.

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